%% Books
@book{fds318258,
Author = {R. Bryant and Bryant, RL and Chern, SS and Gardner, RB and Goldschmidt, HL and Griffiths, PA},
Title = {Exterior Differential Systems},
Pages = {475 pages},
Publisher = {Springer},
Year = {2011},
Month = {December},
ISBN = {9781461397168},
MRNUMBER = {92h:58007},
url = {http://www.ams.org/mathscinet-getitem?mr=92h:58007},
Abstract = {This book gives a treatment of exterior differential
systems.},
Key = {fds318258}
}
@book{fds318259,
Author = {R. Bryant and Gu, C and Berger, M and Bryant, RL},
Title = {Differential Geometry and Differential Equations Proceedings
of a Symposium, held in Shanghai, June 21 - July 6,
1985},
Pages = {246 pages},
Publisher = {Springer},
Year = {2006},
Month = {November},
ISBN = {9783540478836},
Abstract = {The DD6 Symposium was, like its predecessors DD1 to DD5 both
a research symposium and a summer seminar and concentrated
on differential geometry. This volume contains a selection
of the invited papers and some additional
contributions.},
Key = {fds318259}
}
@book{fds318267,
Author = {R. Bryant and Bao, DD-W},
Title = {A Sampler of Riemann-Finsler Geometry},
Volume = {50},
Series = {Mathematical Sciences Research Institute
Publications},
Pages = {363 pages},
Publisher = {Cambridge University Press},
Editor = {Bao, D and Bryant, RL and Chern, S-S and Shen, Z},
Year = {2004},
Month = {November},
ISBN = {9780521831819},
MRNUMBER = {MR2132655(2005j:53003)},
url = {http://www.ams.org/mathscinet-getitem?mr=2132655},
Abstract = {These expository accounts treat issues related to volume,
geodesics, curvature and mathematical biology, with
instructive examples.},
Key = {fds318267}
}
@book{fds318268,
Author = {R. Bryant and Bryant, R and Griffiths, P and Grossman, D},
Title = {Exterior Differential Systems and Euler-Lagrange Partial
Differential Equations},
Series = {Chicago Lectures in Mathematics},
Pages = {213 pages},
Publisher = {University of Chicago Press},
Year = {2003},
Month = {July},
ISBN = {9780226077932},
MRNUMBER = {MR1985469},
url = {http://arxiv.org/abs/math/0207039},
Abstract = {The book also covers the Second Variation, Euler-Lagrange
PDE systems, and higher-order conservation
laws.},
Key = {fds318268}
}
@book{fds43013,
Title = {Selected works of Phillip A. Griffiths with commentary. Part
4. Differential systems.},
Publisher = {American Mathematical Society, Providence, RI; International
Press, Somerville, MA},
Editor = {R. L. Bryant and David R. Morrison},
Year = {2003},
MRNUMBER = {2005e:01025d},
url = {http://www.ams.org/mathscinet-getitem?mr=2005e:01025d},
Key = {fds43013}
}
@book{fds318279,
Author = {R. Bryant and Victor Guillemin and Sigurdur Helgason and R. O. Wells, Jr.},
Title = {Integral Geometry},
Volume = {63},
Pages = {350 pages},
Publisher = {American Mathematical Society},
Editor = {Bryant, R and Guillemin, V and Helgason, S and Wells,
RO},
Year = {1987},
ISBN = {0-8218-5071-7},
MRNUMBER = {87j:53003},
url = {http://www.ams.org/mathscinet-getitem?mr=87j:53003},
Abstract = {Proceedings of the AMS-IMS-SIAM joint summer research
conference held in Brunswick, Maine, August 12–18,
1984},
Key = {fds318279}
}
@book{fds10113,
Author = {R. Bryant and Marcel Berger and Chao Hao Gu},
Title = {Differential Geometry and Differential Equations},
Journal = {Proceedings of the sixth symposium held at Fudan University,
Shanghai, June 21--July 6, 1985, pp. xii+243, 1987,
Springer-Verlag, Berlin},
MRNUMBER = {88b:53002},
url = {http://www.ams.org/mathscinet-getitem?mr=88b:53002},
Key = {fds10113}
}
%% Papers Published
@article{fds375372,
Author = {Bryant, R},
Title = {The generality of closed G_2 solitons},
Journal = {Pure and Applied Mathematics Quarterly},
Volume = {19},
Number = {6},
Pages = {2827-2840},
Publisher = {International Press},
Editor = {Cheng, S-Y and Lima-Filho, P and Yau, SS-T and Yau,
S-T},
Year = {2024},
Month = {January},
Abstract = {The local generality of the space of solitons for the
Laplacian flow of closed G2-structures is analyzed, and it
is shown that the germs of such structures depend, up to
diffeomorphism, on 16 functions of 6 variables (in the sense
of É. Cartan). The method is to construct a natural
exterior differential system whose integral manifolds
describe such solitons and to show that it is involutive in
Cartan’s sense, so that Cartan-Kähler theory can be
applied. Meanwhile, it turns out that, for the more special
case of gradient solitons, the natural exterior differential
system is not involutive, and the generality of these
structures remains a mystery.},
Key = {fds375372}
}
@article{fds375373,
Author = {Bryant, R and Cheeger, J and Lima-Filho, P and Rosenberg, J and White,
B},
Title = {The mathematical work of H. Blaine Lawson,
Jr.},
Journal = {Pure and Applied Mathematics Quarterly},
Volume = {19},
Number = {6},
Pages = {2627-2662},
Publisher = {International Press},
Year = {2024},
Month = {January},
Key = {fds375373}
}
@article{fds368010,
Author = {Bryant, R and Cheeger, J and Griffiths, P and Blum, L and Burns, D and Connes, A and Donnelly, H and Ebin, D and Guillemin, V and Palais, R and Rossi, H and Simons, J and Singer, E and Singer, N and Stanton, N and Sternberg, S},
Title = {Isadore M. Singer (1924–2021) In Memoriam Part 2: Personal
Recollections},
Journal = {Notices of the American Mathematical Society},
Volume = {69},
Number = {10},
Pages = {1-1},
Publisher = {American Mathematical Society (AMS)},
Year = {2022},
Month = {November},
url = {http://dx.doi.org/10.1090/noti2573},
Doi = {10.1090/noti2573},
Key = {fds368010}
}
@article{fds368011,
Author = {Bryant, R and Bismut, J-M and Cheeger, J and Griffiths, P and Donaldson,
S and Hitchin, N and Lawson, HB and Gromov, M and Marcus, A and Spielman,
D and Srivastava, N and Witten, E},
Title = {Isadore M. Singer (1924–2021) In Memoriam Part 1:
Scientific Works},
Journal = {Notices of the American Mathematical Society},
Volume = {69},
Number = {09},
Pages = {1-1},
Publisher = {American Mathematical Society (AMS)},
Year = {2022},
Month = {October},
url = {http://dx.doi.org/10.1090/noti2546},
Doi = {10.1090/noti2546},
Key = {fds368011}
}
@article{fds368012,
Author = {Phong, DH and Siu, Y-T and Bryant, R and Chau, A and Falbel, E and Fefferman, C and Friedman, R and Morgan, J and Futaki, A and Griffiths,
P and Kohn, JJ and Mok, N and Mori, S and Namba, M and Noguchi, J and Ohsawa,
T and Sato, M and Yau, S-T},
Title = {Masatake Kuranishi (1924–2021)},
Journal = {Notices of the American Mathematical Society},
Volume = {69},
Number = {05},
Pages = {1-1},
Publisher = {American Mathematical Society (AMS)},
Year = {2022},
Month = {May},
url = {http://dx.doi.org/10.1090/noti2480},
Doi = {10.1090/noti2480},
Key = {fds368012}
}
@article{fds320296,
Author = {Bryant, RL},
Title = {S.-S. Chern’s study of almost-complex structures on the
six-sphere},
Journal = {International Journal of Mathematics},
Volume = {32},
Number = {12},
Publisher = {World Scientific Publishing},
Editor = {Bryant, R and Cheng, SY and Griffiths, P and Ma, X and Ni, L and Wallach,
N},
Year = {2021},
Month = {November},
url = {http://dx.doi.org/10.1142/S0129167X21400061},
Keywords = {6-sphere • complex structure • exceptional
geometry},
Abstract = {In April 2003, Chern began a study of almost-complex
structures on the six-sphere, with the idea of exploiting
the special properties of its well-known almost-complex
structure invariant under the exceptional group G2. While he
did not solve the (currently still open) problem of
determining whether there exists an integrable
almost-complex structure on S6, he did prove a significant
identity that resolves the question for an interesting class
of almost-complex structures on S6},
Doi = {10.1142/S0129167X21400061},
Key = {fds320296}
}
@article{fds355195,
Author = {Bryant, RL and Foulon, P and Ivanov, SV and Matveev, VS and Ziller,
W},
Title = {Geodesic behavior for Finsler metrics of constant positive
flag curvature on S2},
Journal = {Journal of Differential Geometry},
Volume = {117},
Number = {1},
Pages = {1-22},
Year = {2021},
Month = {January},
url = {http://dx.doi.org/10.4310/JDG/1609902015},
Abstract = {We study non-reversible Finsler metrics with constant flag
curvature 1 on S2 and show that the geodesic flow of every
such metric is conjugate to that of one of Katok's examples,
which form a 1- parameter family. In particular, the length
of the shortest closed geodesic is a complete invariant of
the geodesic flow. We also show, in any dimension, that the
geodesic flow of a Finsler metric with constant positive
flag curvature is completely integrable. Finally, we give an
example of a Finsler metric on S2with positive flag
curvature such that no two closed geodesics intersect and
show that this is not possible when the metric is reversible
or has constant flag curvature.},
Doi = {10.4310/JDG/1609902015},
Key = {fds355195}
}
@article{fds352300,
Author = {Acharya, BS and Bryant, RL and Salamon, S},
Title = {A circle quotient of a G2 cone},
Journal = {Differential Geometry and its Application},
Volume = {73},
Pages = {43 pages},
Publisher = {Elsevier},
Year = {2020},
Month = {December},
url = {http://dx.doi.org/10.1016/j.difgeo.2020.101681},
Abstract = {A study is made of R6 as a singular quotient of the conical
space R+×CP3 with holonomy G2, with respect to an obvious
action by U(1) on CP3 with fixed points. Closed expressions
are found for the induced metric, and for both the curvature
and symplectic 2-forms characterizing the reduction. All
these tensors are invariant by a diagonal action of SO(3) on
R6, which can be used effectively to describe the resulting
geometrical features.},
Doi = {10.1016/j.difgeo.2020.101681},
Key = {fds352300}
}
@article{fds361584,
Author = {Bryant, RL},
Title = {Notes on spinors in low dimension},
Year = {2020},
Month = {November},
Abstract = {The purpose of these old notes (written in 1998 during a
research project on holonomy of pseudo-Riemannian manifolds
of type (10,1)) is to determine the orbit structure of the
groups Spin(p,q) acting on their spinor spaces for the
values (p,q) = (8,0), (9,0), (9,1), (10,0), (10,1), and
(10,2). I'm making them available on the arXiv because I
continue to get requests for them as well as questions about
how they can be cited.},
Key = {fds361584}
}
@article{fds348659,
Author = {Bryant, RL and Clelland, JN},
Title = {Flat metrics with a prescribed derived coframing},
Journal = {Symmetry, Integrability and Geometry: Methods and
Applications (SIGMA)},
Volume = {16},
Publisher = {National Academy of Science of Ukraine},
Year = {2020},
Month = {January},
url = {http://dx.doi.org/10.3842/SIGMA.2020.004},
Abstract = {The following problem is addressed: A 3-manifold M is
endowed with a triple Ω =(Ω1, Ω2, Ω3) of closed 2-forms.
One wants to construct a coframing ω =(ω1, ω2, ω3) of M
such that, first, dωi = Ωi for i = 1, 2, 3, and, second,
the Riemannian metric g = (ω1)2 + (ω2)2 + (ω3)2 be flat.
We show that, in the ‘nonsingular case’, i.e., when the
three 2-forms Ωip span at least a 2-dimensional subspace of
Λ2(Tp*M) and are real-analytic in some p-centered
coordinates, this problem is always solvable on a
neighborhood of p (Formula Presented) M, with the general
solution ω depending on three arbitrary functions of two
variables. Moreover, the characteristic variety of the
generic solution ω can be taken to be a nonsingular cubic.
Some singular situations are considered as well. In
particular, we show that the problem is solvable locally
when Ω1, Ω2, Ω3 are scalar multiples of a single 2-form
that do not vanish simultaneously and satisfy a
nondegeneracy condition. We also show by example that
solutions may fail to exist when these conditions are not
satisfied.},
Doi = {10.3842/SIGMA.2020.004},
Key = {fds348659}
}
@article{fds320298,
Author = {Bryant, RL and Eastwood, MG and Gover, AR and Neusser,
K},
Title = {Some differential complexes within and beyond parabolic
geometry},
Journal = {Advanced Studies in Pure Mathematics},
Volume = {82},
Number = {Differential Geometry and Tanaka Theory},
Pages = {13-40},
Publisher = {Mathematical Society of Japan},
Year = {2019},
Month = {November},
Abstract = {For smooth manifolds equipped with various geometric
structures, we construct complexes that replace the de Rham
complex in providing an alternative fine resolution of the
sheaf of locally constant functions. In case that the
geometric structure is that of a parabolic geometry, our
complexes coincide with the Bernstein-Gelfand-Gelfand
complex associated with the trivial representation. However,
at least in the cases we discuss, our constructions are
relatively simple and avoid most of the machinery of
parabolic geometry. Moreover, our method extends to certain
geometries beyond the parabolic realm.},
Key = {fds320298}
}
@article{fds345423,
Author = {Bryant, R and Buckmire, R and Khadjavi, L and Lind,
D},
Title = {The origins of spectra, an organization for LGBT
mathematicians},
Journal = {Notices of the American Mathematical Society},
Volume = {66},
Number = {6},
Pages = {875-882},
Year = {2019},
Month = {June},
url = {http://dx.doi.org/10.1090/noti1890},
Doi = {10.1090/noti1890},
Key = {fds345423}
}
@article{fds361535,
Author = {Bryant, RL},
Title = {Notes on Projective, Contact, and Null Curves},
Year = {2019},
Month = {May},
Abstract = {These are notes on some algebraic geometry of complex
projective curves, together with an application to studying
the contact curves in CP^3 and the null curves in the
complex quadric Q^3 in CP^4, related by the well-known Klein
correspondence. Most of this note consists of recounting the
classical background. The main application is the explicit
classification of rational null curves of low degree in Q^3.
I have recently received a number of requests for these
notes, so I am posting them to make them generally
available.},
Key = {fds361535}
}
@article{fds325462,
Author = {Bryant, RL and Huang, L and Mo, X},
Title = {On Finsler surfaces of constant flag curvature with a
Killing field},
Journal = {Journal of Geometry and Physics},
Volume = {116},
Pages = {345-357},
Publisher = {Elsevier BV},
Year = {2017},
Month = {June},
url = {http://dx.doi.org/10.1016/j.geomphys.2017.02.012},
Abstract = {We study two-dimensional Finsler metrics of constant flag
curvature and show that such Finsler metrics that admit a
Killing field can be written in a normal form that depends
on two arbitrary functions of one variable. Furthermore, we
find an approach to calculate these functions for
spherically symmetric Finsler surfaces of constant flag
curvature. In particular, we obtain the normal form of the
Funk metric on the unit disk D2.},
Doi = {10.1016/j.geomphys.2017.02.012},
Key = {fds325462}
}
@article{fds320294,
Author = {Bryant, RL},
Title = {On the Convex Pfaff-Darboux Theorem of Ekeland and
Nirenberg},
Journal = {SIGMA 19},
Volume = {060},
Pages = {10},
Year = {2015},
Month = {December},
url = {http://arxiv.org/abs/1512.07100},
Abstract = {The classical Pfaff-Darboux theorem, which provides local
'normal forms' for $1$-forms on manifolds, has applications
in the theory of certain economic models [Chiappori P.-A.,
Ekeland I., Found. Trends Microecon. 5 (2009), 1-151].
However, the normal forms needed in these models often come
with an additional requirement of some type of convexity,
which is not provided by the classical proofs of the
Pfaff-Darboux theorem. (The appropriate notion of
'convexity' is a feature of the economic model. In the
simplest case, when the economic model is formulated in a
domain in $\mathbb{R}^n$, convexity has its usual meaning.)
In [Methods Appl. Anal. 9 (2002), 329-344], Ekeland and
Nirenberg were able to characterize necessary and sufficient
conditions for a given 1-form $\omega$ to admit a convex
local normal form (and to show that some earlier attempts
[Chiappori P.-A., Ekeland I., Ann. Scuola Norm. Sup. Pisa
Cl. Sci. 4 25 (1997), 287-297] and [Zakalyukin V.M., C. R.
Acad. Sci. Paris S\'er. I Math. 327 (1998), 633-638] at this
characterization had been unsuccessful). In this article,
after providing some necessary background, I prove a
strengthened and generalized convex Pfaff-Darboux theorem,
one that covers the case of a Legendrian foliation in which
the notion of convexity is defined in terms of a
torsion-free affine connection on the underlying manifold.
(The main result of Ekeland and Nirenberg concerns the case
in which the affine connection is flat.)},
Key = {fds320294}
}
@article{fds320295,
Author = {Bryant, RL},
Title = {On the conformal volume of 2-tori},
Year = {2015},
Month = {July},
url = {http://arxiv.org/abs/1507.01485},
Keywords = {conformal volume},
Abstract = {This note provides a proof of a 1985 conjecture of Montiel
and Ros about the conformal volume of tori. (This material
is not really new; I'm making it available now because of
requests related to recent interest in the
conjecture.)},
Key = {fds320295}
}
@article{fds320297,
Author = {Bryant, RL},
Title = {Notes on exterior differential systems},
Year = {2014},
Month = {May},
url = {http://arxiv.org/abs/1405.3116},
Keywords = {exterior differential systems • Lie theory •
differential geometry},
Abstract = {These are notes for a very rapid introduction to the basics
of exterior differential systems and their connection with
what is now known as Lie theory, together with some typical
and not-so-typical applications to illustrate their
use.},
Key = {fds320297}
}
@article{fds361666,
Author = {Bryant, RL},
Title = {Complex analysis and a class of Weingarten
surfaces},
Pages = {10 pages},
Year = {2011},
Month = {May},
Abstract = {An idea of Hopf's for applying complex analysis to the study
of constant mean curvature spheres is generalized to cover a
wider class of spheres, namely, those satisfying a
Weingarten relation of a certain type, namely H = f(H^2-K)
for some smooth function f, where H and K are the mean and
Gauss curvatures, respectively. The results are either not
new or are minor extensions of known results, but the
method, which involves introducing a different conformal
structure on the surface than the one induced by the first
fundamental form, is different from the one used by Hopf and
requires less technical results from the theory of PDE than
Hopf's methods. This is a TeXed version of a manuscript
dating from early 1984. It was never submitted for
publication, though it circulated to some people and has
been referred to from time to time in published articles. It
is being provided now for the convenience of those who have
asked for a copy. Except for the correction of various
grammatical or typographical mistakes and infelicities and
the addition of some (clearly marked) comments at the end of
the introduction, the text is that of the
original.},
Key = {fds361666}
}
@article{fds320299,
Author = {Bryant, R and Xu, F},
Title = {Laplacian Flow for Closed $G_2$-Structures: Short Time
Behavior},
Year = {2011},
Month = {January},
Abstract = {We prove short time existence and uniqueness of solutions to
the Laplacian flow for closed $G_2$ structures on a compact
manifold $M^7$. The result was claimed in \cite{BryantG2},
but its proof has never appeared.},
Key = {fds320299}
}
@article{fds243372,
Author = {Bryant, RL},
Title = {Non‐Embedding and Non‐Extension Results in Special
Holonomy},
Pages = {346-367},
Booktitle = {The Many Facets of Geometry},
Publisher = {Oxford University Press},
Address = {Oxford},
Editor = {Garcia-Prada, O and Bourguignon, JP and Salamon,
S},
Year = {2010},
Month = {Fall},
ISBN = {0199534926},
MRCLASS = {53C29},
MRNUMBER = {MR2681703},
url = {http://dx.doi.org/10.1093/acprof:oso/9780199534920.003.0017},
Abstract = {© Oxford University Press 2010. All rights reserved.In the
early analyses of metrics with special holonomy in
dimensions 7 and 8, particularly in regards to their
existence and generality, heavy use was made of the
Cartan-Kähler theorem, essentially because the analyses
were reduced to the study of overdetermined PDE systems
whose natures were complicated by their diffeomorphism
invariance. The Cartan-Kähler theory is well suited for the
study of such systems and the local properties of their
solutions. However, the Cartan-Kähler theory is not
particularly well suited for studies of global problems for
two reasons: first, it is an approach to PDE that relies
entirely on the local solvability of initial value problems
and, second, the Cartan-Kähler theory is only applicable in
the real-analytic category. Nevertheless, when there are no
other adequate methods for analyzing the local generality of
such systems, the Cartan-Kähler theory is a useful tool and
it has the effect of focusing attention on the initial value
problem as an interesting problem in its own right. This
chapter clarifies some of the existence issues involved in
applying the initial value problem to the problem of
constructing metrics with special holonomy. In particular,
it discusses the role of the assumption of real-analyticity
and presents examples to show that one cannot generally
avoid such assumptions in the initial value formulations of
these problems.},
Doi = {10.1093/acprof:oso/9780199534920.003.0017},
Key = {fds243372}
}
@article{fds243377,
Author = {Bryant, RL},
Title = {Commentary},
Journal = {Bulletin of the American Mathematical Society},
Volume = {46},
Number = {2},
Pages = {177-178},
Publisher = {American Mathematical Society (AMS)},
Year = {2009},
Month = {April},
ISSN = {0273-0979},
url = {http://dx.doi.org/10.1090/S0273-0979-09-01248-8},
Doi = {10.1090/S0273-0979-09-01248-8},
Key = {fds243377}
}
@article{fds243378,
Author = {R. Bryant and Bryant, R and Dunajski, M and Eastwood, M},
Title = {Metrisability of two-dimensional projective
structures},
Journal = {Journal of Differential Geometry},
Volume = {83},
Number = {3},
Pages = {465-500},
Publisher = {International Press of Boston},
Year = {2009},
Month = {January},
ISSN = {0022-040X},
MRCLASS = {53},
MRNUMBER = {MR2581355},
url = {http://dx.doi.org/10.4310/jdg/1264601033},
Abstract = {We carry out the programme of R. Liouville [19] to construct
an explicit local obstruction to the existence of a
Levi-Civita connection within a given projective structure
[Γ] on a surface. The obstruction is of order 5 in the
components of a connection in a projective class. It can be
expressed as a point invariant for a second order ODE whose
integral curves are the geodesics of [Γ] or as a weighted
scalar projective invariant of the projective class. If the
obstruction vanishes we find the sufficient conditions for
the existence of a metric in the real analytic case. In the
generic case they are expressed by the vanishing of two
invariants of order 6 in the connection. In degenerate cases
the sufficient obstruction is of order at most 8. © 2009 J.
differential geometry.},
Doi = {10.4310/jdg/1264601033},
Key = {fds243378}
}
@article{fds320198,
Author = {Bryant, RL},
Title = {GRADIENT KAHLER RICCI SOLITONS},
Journal = {ASTERISQUE},
Volume = {321},
Series = {Astérisque},
Number = {321},
Pages = {51-97},
Booktitle = {Géométrie différentielle, physique mathématique,
mathématiques et société. I.},
Publisher = {SOC MATHEMATIQUE FRANCE},
Year = {2008},
Month = {Spring},
ISBN = {978-285629-258-7},
MRCLASS = {53C55 (53C21)},
MRNUMBER = {2010i:53138},
url = {http://arxiv.org/abs/math/0407453},
Abstract = {Some observations about the local and global generality of
gradient Kahler Ricci solitons are made, including the
existence of a canonically associated holomorphic volume
form and vector field, the local generality of solutions
with a prescribed holomorphic volume form and vector field,
and the existence of Poincaré coordinates in the case that
the Ricci curvature is positive and the vector field has a
fixed point. © Asterisque 321.},
Key = {fds320198}
}
@article{fds243385,
Author = {R. Bryant and Bryant, RL and Manno, G and Matveev, VS},
Title = {A solution of a problem of Sophus Lie: Normal forms of
two-dimensional metrics admitting two projective vector
fields},
Journal = {Mathematische Annalen},
Volume = {340},
Number = {2},
Pages = {437-463},
Publisher = {Springer Nature America, Inc},
Year = {2008},
Month = {Spring},
url = {http://www.arxiv.org/abs/0705.3592},
Abstract = {We give a complete list of normal forms for the
two-dimensional metrics that admit a transitive Lie
pseudogroup of geodesic-preserving transformations and we
show that these normal forms are mutually non-isometric.
This solves a problem posed by Sophus Lie. © 2007
Springer-Verlag.},
Doi = {10.1007/s00208-007-0158-3},
Key = {fds243385}
}
@article{fds318264,
Author = {Bryant, RL},
Title = {GEODESICALLY REVERSIBLE FINSLER 2-SPHERES OF CONSTANT
CURVATURE},
Volume = {11},
Series = {Nankai Tracts in Mathematics},
Pages = {95-111},
Booktitle = {Nankai Tracts in Mathematics},
Publisher = {WORLD SCIENTIFIC},
Editor = {Griffiths, PA},
Year = {2006},
Month = {Winter},
ISBN = {9789812700612},
url = {http://dx.doi.org/10.1142/9789812772688_0004},
Abstract = {A Finsler space is said to be geodesically reversible if
each oriented geodesic can be reparametrized as a geodesic
with the reverse orientation. A reversible Finsler space is
geodesically reversible, but the converse need not be true.
In this note, building on recent work of LeBrun and Mason,
it is shown that a geodesically reversible Finsler metric of
constant flag curvature on the 2-sphere is necessarily
projectively flat. As a corollary, using a previous result
of the author, it is shown that a reversible Finsler metric
of constant flag curvature on the 2-sphere is necessarily a
Riemannian metric of constant Gauss curvature, thus settling
a long- standing problem in Finsler geometry.},
Doi = {10.1142/9789812772688_0004},
Key = {fds318264}
}
@article{fds243386,
Author = {Bryant, RL},
Title = {On the geometry of almost complex 6-manifolds},
Journal = {ASIAN JOURNAL OF MATHEMATICS},
Volume = {10},
Number = {3},
Pages = {561-605},
Publisher = {INT PRESS},
Year = {2006},
Month = {September},
url = {http://dx.doi.org/10.4310/AJM.2006.v10.n3.a4},
Keywords = {almost complex manifolds • quasi-integrable •
Nijenhuis tensor},
Abstract = {This article is mostly a writeup of two talks, the first
given in the Besse Seminar at the Ecole Polytechnique in
1998 and the second given at the 2000 International Congress
on Differential Geometry in memory of Alfred Gray in Bilbao,
Spain. It begins with a discussion of basic geometry of
almost complex 6-manifolds. In particular, I define a 2-
parameter family of intrinsic first-order functionals on
almost complex structures on 6-manifolds and compute their
Euler-Lagrange equations. It also includes a discussion of a
natural generalization of holomorphic bundles over complex
manifolds to the almost complex case. The general almost
complex manifold will not admit any nontrivial bundles of
this type, but there is a large class of nonintegrable
almost complex manifolds for which there are such nontrivial
bundles. For example, the standard almost complex structure
on the 6-sphere admits such nontrivial bundles. This class
of almost complex manifolds in dimension 6 will be referred
to as quasi-integrable. Some of the properties of
quasi-integrable structures (both almost complex and
unitary) are developed and some examples are given. However,
it turns out that quasi-integrability is not an involutive
condition, so the full generality of these structures in
Cartan's sense is not well-understood.},
Doi = {10.4310/AJM.2006.v10.n3.a4},
Key = {fds243386}
}
@article{fds318260,
Author = {Bryant, RL},
Title = {CONFORMAL GEOMETRY AND 3-PLANE FIELDS ON
6-MANIFOLDS(Developments of Cartan Geometry and Related
Mathematical Problems)},
Journal = {数理解析研究所講究録},
Volume = {1502},
Series = {RIMS Symposium Proceedings},
Pages = {1-15},
Booktitle = {Developments of Cartan Geometry and Related Mathematical
Problems},
Publisher = {京都大学},
Year = {2006},
Month = {July},
url = {http://arxiv.org/abs/math/0511110},
Keywords = {differential invariants},
Abstract = {The purpose of this note is to provide yet another example
of the link between certain conformal geometries and
ordinary differential equations, along the lines of the
examples discussed by Nurowski in math.DG/0406400. In this
particular case, I consider the equivalence problem for
3-plane fields D on 6-manifolds M that satisfy the
nondegeneracy condition that D+[D,D]=TM I give a solution of
the equivalence problem for such D (as Tanaka has
previously), showing that it defines a so(4,3)- valued
Cartan connection on a principal right H-bundle over M where
H is the subgroup of SO(4,3) that stabilizes a null 3-plane
in R^{4,3}. Along the way, I observe that there is
associated to each such D a canonical conformal structure of
split type on M, one that depends on two derivatives of the
plane field D. I show how the primary curvature tensor of
the Cartan connection associated to the equivalence problem
for D can be interpreted as the Weyl curvature of the
associated conformal structure and, moreover, show that the
split conformal structures in dimension 6 that arise in this
fashion are exactly the ones whose so(4,4)-valued Cartan
connection admits a reduction to a spin(4,3)-connection. I
also discuss how this case has features that are analogous
to those of Nurowski's examples.},
Key = {fds318260}
}
@article{fds243387,
Author = {Bryant, RL},
Title = {SO(n)-Invariant special lagrangian submanifolds of Cn+1 with
fixed loci},
Journal = {CHINESE ANNALS OF MATHEMATICS SERIES B},
Volume = {27},
Number = {1},
Pages = {95-112},
Publisher = {SHANGHAI SCIENTIFIC TECHNOLOGY LITERATURE PUBLISHING
HOUSE},
Year = {2006},
Month = {January},
MRNUMBER = {MR2209954},
url = {http://dx.doi.org/10.1007/s11401-005-0368-5},
Keywords = {calibrations, special Lagrangian submanifolds},
Abstract = {Let SO(n) act in the standard way on C^n and extend this
action in the usual way to C^{n+1}. It is shown that
nonsingular special Lagrangian submanifold L in C^{n+1} that
is invariant under this SO(n)-action intersects the fixed
line C in a nonsingular real-analytic arc A (that may be
empty). If n>2, then A has no compact component. Conversely,
an embedded, noncompact nonsingular real-analytic arc A in C
lies in an embedded nonsingular special Lagrangian
submanifold that is SO(n)-invariant. The same existence
result holds for compact A if n=2. If A is connected, there
exist n distinct nonsingular SO(n)- invariant special
Lagrangian extensions of A such that any embedded
nonsingular SO(n)-invariant special Lagrangian extension of
A agrees with one of these n extensions in some open
neighborhood of A. The method employed is an analysis of a
singular nonlinear PDE and ultimately calls on the work of
Gerard and Tahara to prove the existence of the
extension.},
Doi = {10.1007/s11401-005-0368-5},
Key = {fds243387}
}
@article{fds318261,
Author = {Bryant, R and Freed, D},
Title = {Shiing-Shen Chern},
Journal = {Physics Today},
Volume = {59},
Number = {1},
Pages = {70-72},
Publisher = {AIP Publishing},
Year = {2006},
Month = {January},
url = {http://dx.doi.org/10.1063/1.2180187},
Doi = {10.1063/1.2180187},
Key = {fds318261}
}
@article{fds318262,
Author = {Bryant, RL},
Title = {Second order families of special Lagrangian
3-folds},
Journal = {Perspectives in Riemannian Geometry},
Volume = {40},
Series = {CRM Proceedings and Lecture Notes},
Pages = {63-98},
Booktitle = {Perspectives in Riemannian Geometry},
Publisher = {AMER MATHEMATICAL SOC},
Editor = {Apostolov, V and Dancer, A and Hitchin, N and Wang,
M},
Year = {2006},
Month = {January},
ISBN = {0-8218-3852-0},
url = {http://arxiv.org/abs/math/0007128},
Abstract = {A second order family of special Lagrangian submanifolds of
complex m-space is a family characterized by the
satisfaction of a set of pointwise conditions on the second
fundamental form. For example, the set of ruled special
Lagrangian submanifolds of complex 3-space is characterized
by a single algebraic equation on the second fundamental
form. While the `generic' set of such conditions turns out
to be incompatible, i.e., there are no special Lagrangian
submanifolds that satisfy them, there are many interesting
sets of conditions for which the corresponding family is
unexpectedly large. In some cases, these geometrically
defined families can be described explicitly, leading to new
examples of special Lagrangian submanifolds. In other cases,
these conditions characterize already known families in a
new way. For example, the examples of Lawlor-Harvey
constructed for the solution of the angle conjecture and
recently generalized by Joyce turn out to be a natural and
easily described second order family.},
Key = {fds318262}
}
@article{fds318263,
Author = {Bryant, RL},
Title = {Geometry of manifolds with special holonomy: "100 years of
holonomy"},
Journal = {150 Years of Mathematics at Washington University in St.
Louis},
Volume = {395},
Series = {Contemporary Mathematics},
Pages = {29-38},
Booktitle = {150 years of mathematics at Washington University in St.
Louis},
Publisher = {AMER MATHEMATICAL SOC},
Editor = {Jensen, GR and Krantz, SG},
Year = {2006},
Month = {January},
ISBN = {0-8218-3603-X},
MRNUMBER = {MR2206889},
url = {http://www.ams.org/mathscinet-getitem?mr=2206889},
Keywords = {53C29 (70F25)},
Key = {fds318263}
}
@article{fds318266,
Author = {Bryant, R},
Title = {Holonomy and Special Geometries},
Series = {Conference Proceedings and Lecture Notes in Geometry and
Topology},
Pages = {71-90},
Booktitle = {Dirac Operators: Yesterday and Today},
Publisher = {International Press},
Editor = {Bourguinon, JP and Branson, T and Chamseddine, A and Hijazi, O and Stanton, R},
Year = {2005},
ISBN = {1-57146-175-2},
MRNUMBER = {MR2205367},
url = {http://www.ams.org/mathscinet-getitem?mr=2205367},
Key = {fds318266}
}
@article{fds320300,
Author = {Bryant, RL},
Title = {Real hypersurfaces in unimodular complex
surfaces},
Year = {2004},
Month = {July},
url = {http://arxiv.org/abs/math/0407472},
Abstract = {A unimodular complex surface is a complex 2-manifold X
endowed with a holomorphic volume form. A strictly
pseudoconvex real hypersurface M in X inherits not only a
CR-structure but a canonical coframing as well. In this
article, this canonical coframing on M is defined, its
invariants are discussed and interpreted geometrically, and
its basic properties are studied. A natural evolution
equation for strictly pseudoconvex real hypersurfaces in
unimodular complex surfaces is defined, some of its
properties are discussed, and several examples are computed.
The locally homogeneous examples are determined and used to
illustrate various features of the geometry of the induced
structure on the hypersurface.},
Key = {fds320300}
}
@article{fds243379,
Author = {R. Bryant and Bryant, R and Edelsbrunner, H and Koehl, P and Levitt,
M},
Title = {The area derivative of a space-filling diagram},
Journal = {Discrete and Computational Geometry},
Volume = {32},
Number = {3},
Pages = {293-308},
Publisher = {Springer Nature},
Year = {2004},
Month = {January},
MRNUMBER = {2005k:92077},
url = {http://dx.doi.org/10.1007/s00454-004-1099-1},
Abstract = {The motion of a biomolecule greatly depends on the engulfing
solution, which is mostly water. Instead of representing
individual water molecules, it is desirable to develop
implicit solvent models that nevertheless accurately
represent the contribution of the solvent interaction to the
motion. In such models, hydrophobicity is expressed as a
weighted sum of atomic surface areas. The derivatives of
these weighted areas contribute to the force that drives the
motion. In this paper we give formulas for the weighted and
unweighted area derivatives of a molecule modeled as a
space-filling diagram made up of balls in motion. Other than
the radii and the centers of the balls, the formulas are
given in terms of the sizes of circular arcs of the boundary
and edges of the power diagram. We also give
inclusion-exclusion formulas for these sizes.},
Doi = {10.1007/s00454-004-1099-1},
Key = {fds243379}
}
@article{fds318265,
Author = {Bryant, RL},
Title = {Some remarks on G_2-structures},
Pages = {75-109},
Booktitle = {Proceedings of Gökova Geometry-Topology Conference
2005},
Publisher = {International Press},
Editor = {Akbulut, S and Onder, T and Stern, R},
Year = {2003},
Month = {May},
ISBN = {1-57146-152-3},
url = {http://arxiv.org/abs/math/0305124},
Abstract = {This article consists of some loosely related remarks about
the geometry of G_2-structures on 7-manifolds and is partly
based on old unpublished joint work with two other people:
F. Reese Harvey and Steven Altschuler. Much of this work has
since been subsumed in the work of Hitchin
\cite{MR02m:53070} and Joyce \cite{MR01k:53093}. I am making
it available now mainly because of interest expressed by
others in seeing these results written up since they do not
seem to have all made it into the literature. A formula is
derived for the scalar curvature and Ricci curvature of a
G_2-structure in terms of its torsion. When the fundamental
3-form of the G_2-structure is closed, this formula implies,
in particular, that the scalar curvature of the underlying
metric is nonpositive and vanishes if and only if the
structure is torsion-free. This version contains some new
results on the pinching of Ricci curvature for metrics
associated to closed G_2-structures. Some formulae are
derived for closed solutions of the Laplacian flow that
specify how various related quantities, such as the torsion
and the metric, evolve with the flow. These may be useful in
studying convergence or long-time existence for given
initial data.},
Key = {fds318265}
}
@article{fds243380,
Author = {Bryant, RL},
Title = {Some remarks on Finsler manifolds with constant flag
curvature},
Journal = {HOUSTON JOURNAL OF MATHEMATICS},
Volume = {28},
Number = {2},
Pages = {221-262},
Publisher = {UNIV HOUSTON},
Year = {2002},
Month = {January},
MRNUMBER = {2003h:53102},
url = {HJM},
Abstract = {This article is an exposition of four loosely related
remarks on the geometry of Finsler manifolds with constant
positive flag curvature. The first remark is that there is a
canonical Kahler structure on the space of geodesics of such
a manifold. The second remark is that there is a natural way
to construct a (not necessarily complete) Finsler n-manifold
of constant positive flag curvature out of a hypersurface in
suitably general position in complex projective n-space. The
third remark is that there is a description of the Finsler
metrics of constant curvature on the 2-sphere in terms of a
Riemannian metric and 1-form on the space of its geodesics.
In particular, this allows one to use any (Riemannian) Zoll
metric of positive Gauss curvature on the 2-sphere to
construct a global Finsler metric of constant positive
curvature on the 2-sphere. The fourth remark concerns the
generality of the space of (local) Finsler metrics of
constant positive flag curvature in dimension n+1>2 . It is
shown that such metrics depend on n(n+1) arbitrary functions
of n+1 variables and that such metrics naturally correspond
to certain torsion- free S^1 x GL(n,R)-structures on
2n-manifolds. As a by- product, it is found that these
groups do occur as the holonomy of torsion-free affine
connections in dimension 2n, a hitherto unsuspected
phenomenon. },
Key = {fds243380}
}
@article{fds10364,
Title = {Levi-flat minimal hypersurfaces in two-dimensional complex
space forms},
Volume = {37},
Series = {Adv. Stud. Pure Math.},
Pages = {1--44},
Booktitle = {Lie groups, geometric structures and differential
equations---one hundred years after Sophus Lie (Kyoto/Nara,
1999)},
Publisher = {Math. Soc. Japan},
Year = {2002},
MRNUMBER = {MR1980895},
url = {http://arxiv.org/abs/math/9909159},
Abstract = {The purpose of this article is to classify the real
hypersurfaces in complex space forms of dimension 2 that are
both Levi-flat and minimal. The main results are as follows:
When the curvature of the complex space form is nonzero,
there is a 1-parameter family of such hypersurfaces.
Specifically, for each one-parameter subgroup of the
isometry group of the complex space form, there is an
essentially unique example that is invariant under this
one-parameter subgroup. On the other hand, when the
curvature of the space form is zero, i.e., when the space
form is complex 2-space with its standard flat metric, there
is an additional `exceptional' example that has no
continuous symmetries but is invariant under a lattice of
translations. Up to isometry and homothety, this is the
unique example with no continuous symmetries.},
Key = {fds10364}
}
@article{MR2002i:53010,
Author = {Bryant, RI},
Title = {On Surfaces with Prescribed Shape Operator},
Journal = {Results in Mathematics},
Volume = {40},
Number = {1-4},
Pages = {88-121},
Publisher = {Springer Science and Business Media LLC},
Year = {2001},
Month = {October},
MRNUMBER = {2002i:53010},
url = {http://dx.doi.org/10.1007/bf03322701},
Abstract = {The problem of immersing a simply connected surface with a
prescribed shape operator is discussed. It is shown that,
aside from some special degenerate cases, such as when the
shape operator can be realized by a surface with one family
of principal curves being geodesic, the space of such
realizations is a convex set in an affine space of dimension
at most 3. The cases where this maximum dimension of
realizability is achieved are analyzed and it is found that
there are two such families of shape operators, one
depending essentially on three arbitrary functions of one
variable and another depending essentially on two arbitrary
functions of one variable. The space of realizations is
discussed in each case, along with some of their remarkable
geometric properties. Several explicit examples are
constructed.},
Doi = {10.1007/bf03322701},
Key = {MR2002i:53010}
}
@article{fds243382,
Author = {Bryant, RL},
Title = {Bochner-Kahler metrics},
Journal = {JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY},
Volume = {14},
Number = {3},
Pages = {623-715},
Publisher = {AMER MATHEMATICAL SOC},
Year = {2001},
Month = {January},
MRNUMBER = {2002i:53096},
url = {http://dx.doi.org/10.1090/S0894-0347-01-00366-6},
Abstract = {A Kahler metric is said to be Bochner-Kahler if its Bochner
curvature vanishes. This is a nontrivial condition when the
complex dimension of the underlying manifold is at least 2.
In this article it will be shown that, in a certain well-
defined sense, the space of Bochner-Kahler metrics in
complex dimension n has real dimension n+1 and a recipe for
an explicit formula for any Bochner-Kahler metric is given.
It is shown that any Bochner-Kahler metric in complex
dimension n has local (real) cohomogeneity at most~n. The
Bochner-Kahler metrics that can be `analytically continued'
to a complete metric, free of singularities, are identified.
In particular, it is shown that the only compact Bochner-
Kahler manifolds are the discrete quotients of the known
symmetric examples. However, there are compact Bochner-
Kahler orbifolds that are not locally symmetric. In fact,
every weighted projective space carries a Bochner-Kahler
metric. The fundamental technique is to construct a
canonical infinitesimal torus action on a Bochner-Kahler
metric whose associated momentum mapping has the orbits of
its symmetry pseudo-groupoid as fibers.},
Doi = {10.1090/S0894-0347-01-00366-6},
Key = {fds243382}
}
@article{fds243409,
Author = {Bryant, RL},
Title = {Calibrated embeddings in the special Lagrangian and
coassociative cases},
Journal = {ANNALS OF GLOBAL ANALYSIS AND GEOMETRY},
Volume = {18},
Number = {3-4},
Pages = {405-435},
Publisher = {SPRINGER},
Year = {2000},
Month = {August},
MRNUMBER = {2002j:53063},
url = {http://dx.doi.org/10.1023/A:1006780703789},
Abstract = {Every closed, oriented, real analytic Riemannian 3-manifold
can be isometrically embedded as a special Lagrangian
submanifold of a Calabi-Yau 3-fold, even as the real locus
of an antiholomorphic, isometric involution. Every closed,
oriented, real analytic Riemannian 4-manifold whose bundle
of self-dual 2-forms is trivial can be isometrically
embedded as a coassociative submanifold in a G_2-manifold,
even as the fixed locus of an anti-G_2 involution. These
results, when coupled with McLean's analysis of the moduli
spaces of such calibrated submanifolds, yield a plentiful
supply of examples of compact calibrated submanifolds with
nontrivial deformation spaces.},
Doi = {10.1023/A:1006780703789},
Key = {fds243409}
}
@article{fds320301,
Author = {Bryant, RL},
Title = {Rigidity and quasi-rigidity of extremal cycles in Hermitian
symmetric spaces},
Year = {2000},
Month = {June},
url = {http://arxiv.org/abs/math/0006186},
Abstract = {I use local differential geometric techniques to prove that
the algebraic cycles in certain extremal homology classes in
Hermitian symmetric spaces are either rigid (i.e.,
deformable only by ambient motions) or quasi-rigid (roughly
speaking, foliated by rigid subvarieties in a nontrivial
way). These rigidity results have a number of applications:
First, they prove that many subvarieties in Grassmannians
and other Hermitian symmetric spaces cannot be smoothed
(i.e., are not homologous to a smooth subvariety). Second,
they provide characterizations of holomorphic bundles over
compact Kahler manifolds that are generated by their global
sections but that have certain polynomials in their Chern
classes vanish (for example, c_2 = 0, c_1c_2 - c_3 = 0, c_3
= 0, etc.).},
Key = {fds320301}
}
@article{fds318270,
Author = {Bryant, RL},
Title = {Pseudo-Riemannian metrics with parallel spinor fields and
vanishing Ricci tensor},
Volume = {4},
Series = {Séminaires & Congrès},
Pages = {53-94},
Booktitle = {Global analysis and harmonic analysis (Marseille-Luminy,
1999)},
Publisher = {Société Mathématique de France},
Editor = {Bourguinon, JP and Branson, T and Hijazi, O},
Year = {2000},
Month = {April},
ISBN = {2-85629-094-9},
MRNUMBER = {2002h:53082},
url = {http://arxiv.org/abs/math/0004073},
Abstract = {I discuss geometry and normal forms for pseudo-Riemannian
metrics with parallel spinor fields in some interesting
dimensions. I also discuss the interaction of these
conditions for parallel spinor fields with the condition
that the Ricci tensor vanish (which, for pseudo-Riemannian
manifolds, is not an automatic consequence of the existence
of a nontrivial parallel spinor field).},
Key = {fds318270}
}
@article{fds243384,
Author = {Bryant, RL},
Title = {Harmonic morphisms with fibers of dimension
one},
Journal = {COMMUNICATIONS IN ANALYSIS AND GEOMETRY},
Volume = {8},
Number = {2},
Pages = {219-265},
Publisher = {INT PRESS CO LTD},
Year = {2000},
Month = {April},
MRNUMBER = {2001i:53101},
url = {http://dx.doi.org/10.4310/CAG.2000.v8.n2.a1},
Abstract = {I prove three classification results about harmonic
morphisms whose fibers have dimension one. All are valid
when the domain is at least of dimension 4. (The character
of this overdetermined problem is very different when the
dimension of the domain is 3 or less.) The first result is a
local classification for such harmonic morphisms with
specified target metric, the second is a finiteness theorem
for such harmonic morphisms with specified domain metric,
and the third is a complete classification of such harmonic
morphisms when the domain is a space form of constant
sectional curvature. The methods used are exterior
differential systems and the moving frame. The basic results
are local, but, because of the rigidity of the solutions,
they allow a complete global classification.},
Doi = {10.4310/CAG.2000.v8.n2.a1},
Key = {fds243384}
}
@article{fds243383,
Author = {Bryant, R},
Title = {Recent advances in the theory of holonomy},
Journal = {ASTERISQUE},
Volume = {266},
Number = {266},
Pages = {351-+},
Publisher = {SOC MATHEMATIQUE FRANCE},
Year = {2000},
Month = {January},
MRNUMBER = {2001h:53067},
url = {http://www.dmi.ens.fr/bourbaki/Prog_juin99.html},
Abstract = {After its introduction by Élie Cartan, the notion of
holonomy has become increasingly important in Riemannian and
affine geometry. Beginning with the fundamental work of
Marcel Berger, the classification of possible holonomy
groups of torsion free connections, either Riemannian or
affine, has continued to be developed, with major
breakthroughs in the last ten years. I will report on the
local classification in the affine case, Joyce's fundamental
work on compact manifolds with exceptional holonomies and
their associated geometries, and some new work on the
classification of holonomies of connections with restricted
torsion, which has recently become of interest in string
theory.},
Key = {fds243383}
}
@article{fds318269,
Author = {Bryant, R},
Title = {Élie Cartan and geometric duality},
Journal = {Journées Élie Cartan 1998 et 1999},
Volume = {16},
Pages = {5-20},
Booktitle = {Journées Élie Cartan 1998 et 1999},
Publisher = {Institut Élie Cartan},
Year = {2000},
url = {http://www.math.duke.edu/~bryant/Cartan.pdf},
Key = {fds318269}
}
@article{fds361667,
Author = {Bryant, RL},
Title = {Recent Advances in the Theory of Holonomy},
Journal = {Seminaire Bourbaki},
Volume = {1998},
Number = {99},
Pages = {351-374},
Year = {1999},
Month = {October},
Abstract = {This article is a report on the status of the problem of
classifying the irriducibly acting subgroups of GL(n,R) that
can appear as the holonomy of a torsion-free affine
connection. In particular, it contains an account of the
completion of the classification of these groups by Chi,
Merkulov, and Schwachhofer as well as of the exterior
differential systems analysis that shows that all of these
groups do, in fact, occur. Some discussion of the results of
Joyce on the existence of compact examples with holonomy G_2
or Spin(7) is also included.},
Key = {fds361667}
}
@article{fds362569,
Author = {Bryant, RL},
Title = {Levi-flat Minimal Hypersurfaces in Two-dimensional Complex
Space Forms},
Journal = {Adv. Stud. Pure Math., 37, Math. Soc. Japan, Tokyo, 2002,
1--44},
Year = {1999},
Month = {September},
Abstract = {The purpose of this article is to classify the real
hypersurfaces in complex space forms of dimension 2 that are
both Levi-flat and minimal. The main results are as follows:
When the curvature of the complex space form is nonzero,
there is a 1-parameter family of such hypersurfaces.
Specifically, for each one-parameter subgroup of the
isometry group of the complex space form, there is an
essentially unique example that is invariant under this
one-parameter subgroup. On the other hand, when the
curvature of the space form is zero, i.e., when the space
form is complex 2-space with its standard flat metric, there
is an additional `exceptional' example that has no
continuous symmetries but is invariant under a lattice of
translations. Up to isometry and homothety, this is the
unique example with no continuous symmetries.},
Key = {fds362569}
}
@article{fds243402,
Author = {Bryant, RL},
Title = {Some examples of special Lagrangian tori},
Journal = {Advances in Theoretical and Mathematical
Physics},
Volume = {3},
Number = {1},
Pages = {83-90},
Publisher = {International Press of Boston},
Year = {1999},
MRNUMBER = {2000f:32033},
url = {http://dx.doi.org/10.4310/atmp.1999.v3.n1.a2},
Abstract = {A short paper giving some examples of smooth hypersurfaces M
of degree n+1 in complex projective n-space that are defined
by real polynomial equations and whose real slice contains a
component diffeomorphic to an n-1 torus, which is then
special Lagrangian with respect to the Calabi-Yau metric on
M.},
Doi = {10.4310/atmp.1999.v3.n1.a2},
Key = {fds243402}
}
@article{fds243408,
Author = {R. Bryant and Sharpe, E and Bryant, RL},
Title = {D-branes and Spin^c-structures},
Journal = {Physics Letters, Section B: Nuclear, Elementary Particle and
High-Energy Physics},
Volume = {450},
Number = {4},
Pages = {353-357},
Publisher = {Elsevier BV},
Year = {1999},
MRNUMBER = {2000c:53054},
url = {http://dx.doi.org/10.1016/S0370-2693(99)00161-6},
Abstract = {It was recently pointed out by E. Witten that for a D-brane
to consistently wrap a submanifold of some manifold, the
normal bundle must admit a Spin^c structure. We examine this
constraint in the case of type II string compactifications
with vanishing cosmological constant and argue that, in all
such cases, the normal bundle to a sypersymmetric cycle is
automatically Spin^c.},
Doi = {10.1016/S0370-2693(99)00161-6},
Key = {fds243408}
}
@article{fds10011,
Author = {Russell, Thomas and Farris, Frank},
Title = {Integrability, Gorman systems, and the Lie bracket structure
of the real line (with an appendix by )},
Journal = {J. Math. Econom.},
Volume = {29},
Number = {2},
Pages = {183209},
Year = {1998},
MRNUMBER = {99f:90029},
url = {http://www.ams.org/mathscinet-getitem?mr=99f:90029},
Key = {fds10011}
}
@article{fds243403,
Author = {Bryant, RL},
Title = {Projectively flat Finsler 2-spheres of constant
curvature},
Journal = {Selecta Mathematica},
Volume = {3},
Number = {2},
Pages = {161-203},
Publisher = {Springer Science and Business Media LLC},
Year = {1997},
Month = {March},
MRNUMBER = {98i:53101},
url = {http://dx.doi.org/10.1007/s000290050009},
Abstract = {After recalling the structure equations of Finsler
structures on surfaces, I define a notion of "generalized
Finsler structure" as a way of microlocalizing the problem
of describing Finsler structures subject to curvature
conditions. I then recall the basic notions of path geometry
on a surface and define a notion of "generalized path
geometry" analogous to that of "generalized Finsler
structure." I use these ideas to study the geometry of
Finsler structures on the 2-sphere that have constant
Finsler-Gauss curvature K and whose geodesic path geometry
is projectively flat, i.e., locally equivalent to that of
straight lines in the plane. I show that, modulo
diffeomorphism, there is a 2-parameter family of
projectively flat Finsler structures on the sphere whose
Finsler-Gauss curvature K is identically 1. © Birkhäuser
Verlag, 1997.},
Doi = {10.1007/s000290050009},
Key = {fds243403}
}
@article{fds8915,
Title = {Finsler structures on the 2-sphere satisfying
K=1},
Volume = {196},
Series = {Contemporary Mathematics},
Pages = {2741},
Booktitle = {Finsler geometry (Seattle, WA, 1995)},
Publisher = {Amer. Math. Soc., Providence, RI},
Editor = {David Bao and Shiing-shen Chern and Zhongmin
Shen},
Year = {1996},
MRNUMBER = {97e:53128},
url = {http://www.math.duke.edu/preprints/95-11.dvi},
Key = {fds8915}
}
@article{fds318272,
Author = {Bryant, R},
Title = {Classical, exceptional, and exotic holonomies: a status
report},
Volume = {1},
Series = {Sémin. Congr.},
Pages = {93-165},
Booktitle = {Actes de la Table Ronde de Géométrie Différentielle},
Publisher = {Société Mathématique de France},
Editor = {Besse, A},
Year = {1996},
ISBN = {2-85629-047-7},
MRNUMBER = {98c:53037},
url = {http://www.math.duke.edu/preprints/95-10.dvi},
Abstract = {A survey paper on the status of the holonomy problem as of
1995.},
Key = {fds318272}
}
@article{fds243404,
Author = {R. Bryant and Bryant, R and Griffiths, P and Hsu, L},
Title = {Hyperbolic exterior differential systems and their
conservation laws, part II},
Journal = {Selecta Mathematica, New Series},
Volume = {1},
Number = {2},
Pages = {265-323},
Publisher = {Springer Nature},
Year = {1995},
Month = {September},
MRNUMBER = {97d:580009},
url = {http://www.math.duke.edu/preprints/94-13.dvi},
Doi = {10.1007/BF01671567},
Key = {fds243404}
}
@article{fds369336,
Author = {Bryant, RL and Griffiths, PA},
Title = {Characteristic Cohomology of Differential Systems (I):
General Theory},
Journal = {Journal of the American Mathematical Society},
Volume = {8},
Number = {3},
Pages = {507-507},
Publisher = {JSTOR},
Year = {1995},
Month = {July},
url = {http://dx.doi.org/10.2307/2152923},
Doi = {10.2307/2152923},
Key = {fds369336}
}
@article{fds376556,
Author = {BRYANT, RL and GRIFFITHS, PA},
Title = {CHARACTERISTIC COHOMOLOGY OF DIFFERENTIAL-SYSTEMS .1.
GENERAL-THEORY},
Journal = {JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY},
Volume = {8},
Number = {3},
Pages = {507-596},
Publisher = {AMER MATHEMATICAL SOC},
Year = {1995},
Month = {July},
url = {http://dx.doi.org/10.2307/2152923},
Doi = {10.2307/2152923},
Key = {fds376556}
}
@article{fds243406,
Author = {R. Bryant and BRYANT, RL and GRIFFITHS, PA},
Title = {CHARACTERISTIC COHOMOLOGY OF DIFFERENTIAL-SYSTEMS .2.
CONSERVATION-LAWS FOR A CLASS OF PARABOLIC
EQUATIONS},
Journal = {DUKE MATHEMATICAL JOURNAL},
Volume = {78},
Number = {3},
Pages = {531-676},
Publisher = {DUKE UNIV PRESS},
Year = {1995},
Month = {June},
MRNUMBER = {96d:58158},
url = {http://www.math.duke.edu/preprints/93-02.dvi},
Doi = {10.1215/S0012-7094-95-07824-7},
Key = {fds243406}
}
@article{fds243405,
Author = {R. Bryant and Bryant, R and Griffiths, P and Hsu, L},
Title = {Hyperbolic exterior differential systems and their
conservation laws, part I},
Journal = {Selecta Mathematica, New Series},
Volume = {1},
Number = {1},
Pages = {21-112},
Publisher = {Springer Nature},
Year = {1995},
Month = {March},
MRNUMBER = {97d:580008},
url = {http://www.math.duke.edu/preprints/94-13.dvi},
Doi = {10.1007/BF01614073},
Key = {fds243405}
}
@article{fds318273,
Author = {R. Bryant and BRYANT, R and GRIFFITHS, P and HSU, L},
Title = {TOWARD A GEOMETRY OF DIFFERENTIAL-EQUATIONS},
Journal = {GEOMETRY, TOPOLOGY & PHYSICS},
Volume = {4},
Series = {Conf. Proc. Lecture Notes Geom. Topology},
Pages = {1-76},
Booktitle = {Geometry, Topology, & Physics},
Publisher = {INTERNATIONAL PRESS INC BOSTON},
Editor = {Yau, ST},
Year = {1995},
Month = {January},
ISBN = {1-57146-024-1},
MRNUMBER = {97b:58005},
url = {http://www.math.duke.edu/preprints/94-12.dvi},
Key = {fds318273}
}
@article{fds318274,
Author = {R. Bryant and Bryant, R and Gardner, RB},
Title = {Control Structures},
Volume = {12},
Series = {Banach Center Publications},
Pages = {111-121},
Booktitle = {Geometry in nonlinear control and differential inclusions
(Warsaw, 1993)},
Publisher = {Polish Academy of Sciences},
Editor = {Jakubczyk, B and Respondek, W and Rzezuchowski,
T},
Year = {1995},
MRNUMBER = {96h:93024},
url = {http://www.math.duke.edu/preprints/94-11.dvi},
Key = {fds318274}
}
@article{fds318275,
Author = {Bryant, R},
Title = {An introduction to Lie groups and symplectic
geometry},
Volume = {1},
Series = {IAS/Park City Mathematics},
Pages = {5-181},
Booktitle = {Geometry and quantum field theory (Park City, UT,
1991)},
Publisher = {American Mathematical Society},
Editor = {Freed, D and Uhlenbeck, K},
Year = {1995},
ISBN = {0-8218-0400-6},
MRNUMBER = {96i:58002},
url = {http://www.ams.org/mathscinet-getitem?mr=96i:58002},
Abstract = {A series of lectures on Lie groups and symplectic geometry,
aimed at the beginning graduate student level.},
Key = {fds318275}
}
@article{fds318271,
Author = {Bryant, RL},
Title = {On extremals with prescribed Lagrangian densities},
Volume = {36},
Series = {Symposia Mathematica},
Pages = {86-111},
Booktitle = {Manifolds and geometry (Pisa, 1993)},
Publisher = {Cambridge University Press},
Editor = {Bartolomeis, P and Tricerri, F and Vesentini, E},
Year = {1994},
Month = {June},
ISBN = {0-521-56216-3},
MRNUMBER = {99a:58043},
url = {http://arxiv.org/abs/dg-ga/9406001},
Abstract = {Consider two manifolds~$M^m$ and $N^n$ and a first-order
Lagrangian $L(u)$ for mappings $u:M\to N$, i.e., $L$ is an
expression involving $u$ and its first derivatives whose
value is an $m$-form (or more generally, an $m$-density)
on~$M$. One is usually interested in describing the extrema
of the functional $\Cal L(u) = \int_M L(u)$, and these are
characterized locally as the solutions of the Euler-Lagrange
equation~$E_L(u)=0$ associated to~$L$. In this note I will
discuss three problems which can be understood as trying to
determine how many solutions exist to the Euler-Lagrange
equation which also satisfy $L(u) = \Phi$, where $\Phi$ is a
specified $m$-form or $m$-density on~$M$. The first problem,
which is solved completely, is to determine when two minimal
graphs over a domain in the plane can induce the same area
form without merely differing by a vertical translation or
reflection. The second problem, described more fully below,
arose in Professor Calabi's study of extremal isosystolic
metrics on surfaces. The third problem, also solved
completely, is to determine the (local) harmonic maps
between spheres which have constant energy
density.},
Key = {fds318271}
}
@article{fds243401,
Author = {R. Bryant and Bryant, RL and Hsu, L},
Title = {Rigidity of integral curves of rank 2 distributions},
Journal = {Inventiones Mathematicae},
Volume = {114},
Number = {1},
Pages = {435-461},
Publisher = {Springer Nature},
Year = {1993},
Month = {December},
ISSN = {0020-9910},
MRNUMBER = {94j:58003},
url = {http://www.math.duke.edu/~bryant/Rigid.dvi},
Doi = {10.1007/BF01232676},
Key = {fds243401}
}
@article{fds243400,
Author = {Bryant, RL},
Title = {Some remarks on the geometry of austere manifolds},
Journal = {Boletim da Sociedade Brasileira de Matemática},
Volume = {21},
Number = {2},
Pages = {133-157},
Publisher = {Springer Nature},
Year = {1991},
Month = {September},
MRNUMBER = {92k:53112},
url = {http://www.math.duke.edu/preprints/90-03.dvi},
Abstract = {We prove several structure theorems about the special class
of minimal submanifolds which Harvey and Lawson have called
"austere" and which arose in connection with their
foundational work on calibrations. The condition of
austerity is a pontwise condition on the second fundamental
form and essentially requires that the non-zero eigenvalues
of the second fundamental form in any normal direction at
any point occur in oppositely signed pairs. We solve the
pointwise problem of describing the set of austere second
fundamental forms in dimension at most four and the local
problem of describing the austere three-folds in Euclidean
space in all dimensions. © 1991 Sociedade Brasileira de
Matemática.},
Doi = {10.1007/BF01237361},
Key = {fds243400}
}
@article{fds318276,
Author = {Bryant, R},
Title = {Two exotic holonomies in dimension four, path geometries,
and twistor theory},
Volume = {53},
Series = {Proc. Sympos. Pure Math.},
Pages = {33-88},
Booktitle = {Complex geometry and Lie theory (Sundance, UT,
1989)},
Publisher = {American Mathematical Society},
Editor = {Carlson, J and Clemens, H and Morrison, D},
Year = {1991},
ISBN = {0-8218-1492-3},
MRNUMBER = {93e:53030},
url = {http://www.math.duke.edu/~bryant/ExoticHol.dvi},
Key = {fds318276}
}
@article{fds243398,
Author = {R. Bryant and Bryant, R and Harvey, R},
Title = {Submanifolds in hyper-Kähler geometry},
Journal = {Journal of the American Mathematical Society},
Volume = {2},
Number = {1},
Pages = {1-31},
Publisher = {American Mathematical Society (AMS)},
Year = {1989},
Month = {January},
MRNUMBER = {89m:53090},
url = {http://dx.doi.org/10.1090/S0894-0347-1989-0953169-8},
Doi = {10.1090/S0894-0347-1989-0953169-8},
Key = {fds243398}
}
@article{fds243399,
Author = {R. Bryant and Bryant, RL and Salamon, SM},
Title = {On the construction of some complete metrics with
exceptional holonomy},
Journal = {Duke Mathematical Journal},
Volume = {58},
Number = {3},
Pages = {829-850},
Publisher = {Duke University Press},
Year = {1989},
Month = {January},
MRNUMBER = {90i:53055},
url = {http://dx.doi.org/10.1215/S0012-7094-89-05839-0},
Doi = {10.1215/S0012-7094-89-05839-0},
Key = {fds243399}
}
@article{fds318277,
Author = {Bryant, R},
Title = {Surfaces in conformal geometry},
Volume = {48},
Series = {Proc. Sympos. Pure Math.},
Pages = {227-240},
Booktitle = {The mathematical heritage of Hermann Weyl (Durham, NC,
1987)},
Publisher = {American Mathematical Society},
Editor = {Wells, RO},
Year = {1988},
ISBN = {0-8218-1482-6},
MRNUMBER = {89m:53102},
url = {http://www.ams.org/mathscinet-getitem?mr=89m:53102},
Abstract = {A survey paper. However, there are some new results.
Building on the results in A duality theorm for Willmore
surfaces, I use the Klein correspondance to determine the
moduli space of Willmore critical spheres for low critical
values and also determine the moduli space of Willmore
minima for the real projective plane in 3-space.},
Key = {fds318277}
}
@article{fds318278,
Author = {Bryant, R},
Title = {Surfaces of mean curvature one in hyperbolic
space},
Volume = {154-155},
Series = {Astérisque},
Pages = {321-347},
Booktitle = {Théorie des variétés minimales et applications
(Palaiseau, 1983–1984)},
Publisher = {Société Mathématique de France},
Year = {1988},
MRNUMBER = {955072},
url = {http://www.ams.org/mathscinet-getitem?mr=955072},
Key = {fds318278}
}
@article{fds243397,
Author = {Bryant, RL},
Title = {Metrics with Exceptional Holonomy},
Journal = {The Annals of Mathematics},
Volume = {126},
Number = {3},
Pages = {525-525},
Publisher = {JSTOR},
Year = {1987},
Month = {November},
MRNUMBER = {89b:53084},
url = {http://dx.doi.org/10.2307/1971360},
Doi = {10.2307/1971360},
Key = {fds243397}
}
@article{fds318280,
Author = {Bryant, R},
Title = {On notions of equivalence of variational problems with one
independent variable},
Volume = {68},
Series = {Contemporary Mathematics},
Pages = {65-76},
Booktitle = {Differential geometry: the interface between pure and
applied mathematics (San Antonio, Tex., 1986)},
Publisher = {American Mathematical Society},
Editor = {Luksic, M and Martin, C and Shadwick, W},
Year = {1987},
ISBN = {0-8218-5075-X},
MRNUMBER = {89f:58037},
url = {http://www.ams.org/mathscinet-getitem?mr=89f:58037},
Key = {fds318280}
}
@article{fds318281,
Author = {Bryant, R},
Title = {A survey of Riemannian metrics with special holonomy
groups},
Pages = {505-514},
Booktitle = {Proceedings of the International Congress of Mathematicians.
Vol. 1, 2. (Berkeley, Calif., 1986)},
Publisher = {American Mathematical Society},
Editor = {Gleason, A},
Year = {1987},
ISBN = {0-8218-0110-4},
MRNUMBER = {89f:53068},
url = {http://www.ams.org/mathscinet-getitem?mr=89f:53068},
Key = {fds318281}
}
@article{fds318282,
Author = {Bryant, R},
Title = {Minimal Lagrangian submanifolds of Kähler-Einstein
manifolds},
Volume = {1255},
Series = {Lecture Notes in Math.},
Pages = {1-12},
Booktitle = {Differential geometry and differential equations (Shanghai,
1985)},
Publisher = {Springer-Verlag},
Editor = {Gu, C and Berger, M and Bryant, RL},
Year = {1987},
ISBN = {3-540-17849-X},
MRNUMBER = {88j:53061},
url = {http://www.ams.org/mathscinet-getitem?mr=88j:53061},
Key = {fds318282}
}
@article{fds243396,
Author = {R. Bryant and BRYANT, R and GRIFFITHS, P},
Title = {REDUCTION FOR CONSTRAINED VARIATIONAL-PROBLEMS AND
INTEGRAL-K2/2DS},
Journal = {AMERICAN JOURNAL OF MATHEMATICS},
Volume = {108},
Number = {3},
Pages = {525-570},
Publisher = {JOHNS HOPKINS UNIV PRESS},
Year = {1986},
Month = {June},
MRNUMBER = {88a:58044},
url = {http://dx.doi.org/10.2307/2374654},
Doi = {10.2307/2374654},
Key = {fds243396}
}
@article{fds243394,
Author = {Bryant, RL},
Title = {Minimal surfaces of constant curvature in
sn},
Journal = {Transactions of the American Mathematical
Society},
Volume = {290},
Number = {1},
Pages = {259-271},
Publisher = {JSTOR},
Year = {1985},
Month = {January},
MRNUMBER = {87c:53110},
url = {http://dx.doi.org/10.1090/S0002-9947-1985-0787964-8},
Abstract = {In this note, we study an overdetermined system of partial
differential equations whose solutions determine the minimal
surfaces in Sn of constant Gaussian curvature. If the
Gaussian curvature is positive, the solution to the global
problem was found by [Calabi], while the solution to the
local problem was found by [Wallach]. The case of
nonpositive Gaussian curvature is more subtle and has
remained open. We prove that there are no minimal surfaces
in Sn of constant negative Gaussian curvature (even
locally). We also find all of the flat minimal surfaces in
Sn and give necessary and sufficient conditions that a given
two-torus may be immersed minimally, conformally, and flatly
into Sn. © 1985 American Mathematical Society.},
Doi = {10.1090/S0002-9947-1985-0787964-8},
Key = {fds243394}
}
@article{fds243395,
Author = {Bryant, RL},
Title = {Lie groups and twlstor spaces},
Journal = {Duke Mathematical Journal},
Volume = {52},
Number = {1},
Pages = {223-261},
Publisher = {Duke University Press},
Year = {1985},
Month = {January},
MRNUMBER = {87d:58047},
url = {http://dx.doi.org/10.1215/S0012-7094-85-05213-5},
Doi = {10.1215/S0012-7094-85-05213-5},
Key = {fds243395}
}
@article{fds318283,
Author = {Bryant, R},
Title = {Metrics with holonomy G2 or Spin(7)},
Volume = {1111},
Series = {Lecture Notes in Math.},
Pages = {269-277},
Booktitle = {Workshop Bonn 1984 (Bonn, 1984)},
Publisher = {Springer},
Editor = {Hirzebruch, F and Schwermer, J and Suter, S},
Year = {1985},
MRNUMBER = {87a:53082},
url = {http://www.ams.org/mathscinet-getitem?mr=87a:53082},
Key = {fds318283}
}
@article{fds243393,
Author = {Bryant, RL},
Title = {A duality theorem for willmore surfaces},
Journal = {Journal of Differential Geometry},
Volume = {20},
Number = {1},
Pages = {23-53},
Publisher = {International Press of Boston},
Year = {1984},
Month = {January},
MRNUMBER = {86j:58029},
url = {http://dx.doi.org/10.4310/jdg/1214438991},
Doi = {10.4310/jdg/1214438991},
Key = {fds243393}
}
@article{fds243391,
Author = {R. Bryant and Bryant, RL and Griffiths, PA and Yang, D},
Title = {Characteristics and existence of isometric
embeddings},
Journal = {Duke Mathematical Journal},
Volume = {50},
Number = {4},
Pages = {893-994},
Publisher = {Duke University Press},
Year = {1983},
Month = {January},
MRNUMBER = {85d:53027},
url = {http://dx.doi.org/10.1215/S0012-7094-83-05040-8},
Doi = {10.1215/S0012-7094-83-05040-8},
Key = {fds243391}
}
@article{fds243392,
Author = {R. Bryant and Berger, E and Bryant, R and Griffiths, P},
Title = {The Gauss equations and rigidity of isometric
embeddings},
Journal = {Duke Mathematical Journal},
Volume = {50},
Number = {3},
Pages = {803-892},
Publisher = {Duke University Press},
Year = {1983},
Month = {January},
MRNUMBER = {85k:53056},
url = {http://dx.doi.org/10.1215/S0012-7094-83-05039-1},
Doi = {10.1215/S0012-7094-83-05039-1},
Key = {fds243392}
}
@article{fds318284,
Author = {R. Bryant and Bryant, R and Griffiths, PA},
Title = {Some observations on the infinitesimal period relations for
regular threefolds with trivial canonical
bundle},
Volume = {36},
Series = {Progress in Mathematics},
Pages = {77-102},
Booktitle = {Arithmetic and geometry, Vol. II},
Publisher = {Birkhäuser Boston},
Editor = {Artin, M and Tate, J},
Year = {1983},
ISBN = {3-7643-3133-X},
MRNUMBER = {86a:32044},
url = {http://www.ams.org/mathscinet-getitem?mr=86a:32044},
Key = {fds318284}
}
@article{fds243389,
Author = {Bryant, RL},
Title = {Holomorphic curves in lorentzian cr-manifolds},
Journal = {Transactions of the American Mathematical
Society},
Volume = {272},
Number = {1},
Pages = {203-221},
Publisher = {American Mathematical Society (AMS)},
Year = {1982},
Month = {January},
MRNUMBER = {83i:32029},
url = {http://dx.doi.org/10.1090/S0002-9947-1982-0656486-4},
Abstract = {A CR-manifold is said to be Lorentzian if its Levi form has
one negative eigenvalue and the rest positive. In this case,
it is possible that the CR-manifold contains holomorphic
curves. In this paper, necessary and sufficient conditions
are derived (in terms of the “derivatives” of the
CR-structure) in order that holomorphic curves exist. A
“flatness” theorem is proven characterizing the real
Lorentzian hyperquadric Qs C CP3and examples are given
showing that nonflat Lorentzian hyperquadrics can have a
richer family of holomorphic curves than the flat ones. ©
1982 American Mathematical Society.},
Doi = {10.1090/S0002-9947-1982-0656486-4},
Key = {fds243389}
}
@article{fds243390,
Author = {Bryant, RL},
Title = {Submanifolds and special structures on the
octonians},
Journal = {Journal of Differential Geometry},
Volume = {17},
Number = {2},
Pages = {185-232},
Publisher = {International Press of Boston},
Year = {1982},
Month = {January},
MRNUMBER = {84h:53091},
url = {http://dx.doi.org/10.4310/jdg/1214436919},
Abstract = {A study of the geometry of submanifolds of real 8-space
under the group of motions generated by translations and
rotations in the subgroup Spin(7) instead of the full SO(8).
I call real 8-space endowed with this group O or octonian
space. The fact that the stabilizer of an oriented 2-plane
in Spin(7) is U(3) implies that any oriented 6-manifold in O
inherits a U(3)-structure. The first part of the paper
studies the generality of the 6-manifolds whose inherited
U(3)-structure is symplectic, complex, or Kähler, etc.
by applying the theory of exterior differential systems. I
then turn to the study of the standard 6-sphere in O as an
almost complex manifold and study the space of what are now
called pseudo-holomorphic curves in the 6-sphere. I prove
that every compact Riemann surface occurs as a (possibly
ramified) pseudo-holomorphic curve in the 6-sphere. I also
show that all of the genus zero pseudo-holomorphic curves in
the 6-sphere are algebraic as surfaces. Reprints are
available.},
Doi = {10.4310/jdg/1214436919},
Key = {fds243390}
}
@article{fds243410,
Author = {Bryant, RL},
Title = {Conformal and minimal immersions of compact surfaces into
the 4-sphere},
Journal = {Journal of Differential Geometry},
Volume = {17},
Number = {3},
Pages = {455-473},
Publisher = {International Press of Boston},
Year = {1982},
Month = {January},
MRNUMBER = {84a:53062},
url = {http://dx.doi.org/10.4310/jdg/1214437137},
Doi = {10.4310/jdg/1214437137},
Key = {fds243410}
}
@article{fds318285,
Author = {R. Bryant and Bryant, R and Chern, SS and Griffiths, PA},
Title = {Exterior Differential Systems},
Volume = {1},
Pages = {219-338},
Booktitle = {Proceedings of the 1980 Beijing Symposium on Differential
Geometry and Differential Equations (Beijing,
1980)},
Publisher = {Science Press; Gordon & Breach Science Publishers},
Editor = {Chern, SS and Wu, WT},
Year = {1982},
ISBN = {0-677-16420-3},
MRNUMBER = {85k:58005},
url = {http://www.ams.org/mathscinet-getitem?mr=85k:58005},
Key = {fds318285}
}
@article{fds243388,
Author = {R. Bryant and BERGER, E and BRYANT, R and GRIFFITHS, P},
Title = {SOME ISOMETRIC EMBEDDING AND RIGIDITY RESULTS FOR
RIEMANNIAN-MANIFOLDS},
Journal = {PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE
UNITED STATES OF AMERICA-PHYSICAL SCIENCES},
Volume = {78},
Number = {8},
Pages = {4657-4660},
Year = {1981},
MRNUMBER = {82h:53074},
url = {http://dx.doi.org/10.1073/pnas.78.8.4657},
Doi = {10.1073/pnas.78.8.4657},
Key = {fds243388}
}
%% Papers Accepted
@article{fds216495,
Author = {R. Bryant and Michael G. Eastwood and A. Rod. Gover and Katharina
Neusser},
Title = {Some differential complexes within and beyond parabolic
geometry},
Year = {2011},
Month = {December},
url = {http://arxiv.org/abs/1112.2142v2},
Abstract = {For smooth manifolds equipped with various geometric
structures, we construct complexes that replace the de Rham
complex in providing an alternative fine resolution of the
sheaf of locally constant functions. In case that the
geometric structure is that of a parabolic geometry, our
complexes coincide with the Bernstein- Gelfand-Gelfand
complex associated with the trivial representation. However,
at least in the cases we discuss, our constructions are
relatively simple and avoid most of the machinery of
parabolic geometry. Moreover, our method extends to certain
geometries beyond the parabolic realm.},
Key = {fds216495}
}
%% Preprints
@article{fds225242,
Author = {R.L. Bryant and Feng Xu},
Title = {Laplacian flow for closed G2-structures: short
time behavior},
Year = {2011},
Month = {January},
url = {http://arxiv.org/abs/1101.2004},
Abstract = {We prove short time existence and uniqueness of solutions to
the Laplacian flow for closed G2 structures on a compact
manifold M7. The result was claimed in \cite{BryantG2}, but
its proof has never appeared.},
Key = {fds225242}
}
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