%% 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 = {1461397162},
MRNUMBER = {92h:58007},
url = {http://www.ams.org/mathscinetgetitem?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 = {3540478833},
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 David Bao and S.S. Chern and Zhongmin Shen},
Title = {A Sampler of RiemannFinsler Geometry},
Volume = {50},
Series = {Mathematical Sciences Research Institute
Publications},
Pages = {363 pages},
Publisher = {Cambridge University Press},
Editor = {Bao, D and Bryant, RL and Chern, SS and Shen, Z},
Year = {2004},
Month = {November},
ISBN = {0521831814},
MRNUMBER = {MR2132655(2005j:53003)},
url = {http://www.ams.org/mathscinetgetitem?mr=2132655},
Abstract = {These expository accounts treat issues in Finsler geometry
related to volume, geodesics, curvature and mathematical
biology, with instructive examples.},
Key = {fds318267}
}
@book{fds318268,
Author = {R. Bryant and Bryant, RL and Griffiths, PA and Grossman, DA},
Title = {Exterior Differential Systems and EulerLagrange Partial
Differential Equations},
Series = {Chicago Lectures in Mathematics},
Pages = {213 pages},
Publisher = {University of Chicago Press},
Year = {2003},
Month = {July},
ISBN = {0226077934},
MRNUMBER = {MR1985469},
url = {http://arxiv.org/abs/math/0207039},
Abstract = {We use methods from exterior differential systems (EDS) to
develop a geometric theory of scalar, firstorder Lagrangian
functionals and their associated EulerLagrange PDEs,
subject to contact transformations. The first chapter
contains an introduction of the classical PoincareCartan
form in the context of EDS, followed by proofs of classical
results, including a solution to the relevant inverse
problem, Noether's theorem on symmetries and conservation
laws, and several aspects of minimal hypersurfaces. In the
second chapter, the equivalence problem for PoincareCartan
forms is solved, giving the differential invariants of such
a form, identifying associated geometric structures
(including a family of affine hypersurfaces), and exhibiting
certain "special" EulerLagrange equations characterized by
their invariants. In the third chapter, we discuss a
collection of PoincareCartan forms having a naturally
associated conformal geometry, and exhibit the conservation
laws for nonlinear Poisson and wave equations that result
from this. The fourth and final chapter briefly discusses
additional PDE topics from this viewpointEulerLagrange
PDE systems, higher order Lagrangians and conservation laws,
identification of local minima for Lagrangian functionals,
and Backlund transformations. No previous knowledge of
exterior differential systems or of the calculus of
variations is assumed.},
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/mathscinetgetitem?mr=2005e:01025d},
Key = {fds43013}
}
@book{fds320301,
Author = {Bryant, RL},
Title = {Rigidity and quasirigidity of extremal cycles in Hermitian
symmetric spaces},
Year = {2001},
Month = {March},
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 quasirigid (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}
}
@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 = {0821850717},
MRNUMBER = {87j:53003},
url = {http://www.ams.org/mathscinetgetitem?mr=87j:53003},
Abstract = {Proceedings of the AMSIMSSIAM 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 21July 6, 1985, pp. xii+243, 1987,
SpringerVerlag, Berlin},
MRNUMBER = {88b:53002},
url = {http://www.ams.org/mathscinetgetitem?mr=88b:53002},
Key = {fds10113}
}
%% Papers Published
@article{fds325462,
Author = {Bryant, R 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 = {345357},
Year = {2017},
Month = {June},
url = {http://dx.doi.org/10.1016/j.geomphys.2017.02.012},
Doi = {10.1016/j.geomphys.2017.02.012},
Key = {fds325462}
}
@article{fds320298,
Author = {Bryant, RL and Eastwood, MG and Gover, AR and Neusser,
K},
Title = {Some differential complexes within and beyond parabolic
geometry},
Year = {2012},
Month = {March},
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 BernsteinGelfandGelfand
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{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 = {Nonembedding and nonextension results in special
holonomy},
Pages = {346367},
Booktitle = {The Many Facets of Geometry: A Tribute to Nigel
Hitchin},
Publisher = {Oxford University Press},
Address = {Oxford},
Editor = {GarciaPrada, 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 = {Constructions of metrics with special holonomy by methods of
exterior differential systems are reviewed and the
interpretations of these construction as `flows' on
hypersurface geometries are considered. It is shown that
these hypersurface 'flows' are not generally wellposed for
smooth initial data and counterexamples to existence are
constructed.},
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 = {177178},
Year = {2009},
ISSN = {02730979},
url = {http://dx.doi.org/10.1090/S0273097909012488},
Doi = {10.1090/S0273097909012488},
Key = {fds243377}
}
@article{fds243378,
Author = {R. Bryant and Bryant, RL and Dunajski, M and Eastwood, M},
Title = {Metrisability of twodimensional projective
structures},
Volume = {83},
Number = {3},
Pages = {465499},
Year = {2009},
ISSN = {0022040X},
MRCLASS = {53},
MRNUMBER = {MR2581355},
url = {http://arxiv.org/abs/0801.0300v1},
Abstract = {We carry out the programme of R. Liouville \cite{Liouville}
to construct an explicit local obstruction to the existence
of a LeviCivita connection within a given projective
structure $[\Gamma]$ 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
$[\Gamma]$ 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.},
Key = {fds243378}
}
@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
twodimensional metrics admitting two projective vector
fields},
Journal = {Mathematische Annalen},
Volume = {340},
Number = {2},
Pages = {437463},
Year = {2008},
Month = {Spring},
url = {http://www.arxiv.org/abs/0705.3592},
Abstract = {We give a complete list of normal forms for the
twodimensional metrics that admit a transitive Lie
pseudogroup of geodesicpreserving transformations and we
show that these normal forms are mutually nonisometric.
This solves a problem posed by Sophus Lie. © 2007
SpringerVerlag.},
Doi = {10.1007/s0020800701583},
Key = {fds243385}
}
@article{fds320198,
Author = {Bryant, RL},
Title = {Gradient Kähler Ricci solitons},
Journal = {Asterisque},
Volume = {321},
Series = {Astérisque},
Number = {321},
Pages = {5197},
Booktitle = {Géométrie différentielle, physique mathématique,
mathématiques et société. I.},
Publisher = {Soc. Math. France},
Year = {2008},
Month = {Spring},
ISBN = {9782856292587},
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{fds243386,
Author = {Bryant, RL},
Title = {On the geometry of almost complex 6manifolds},
Journal = {The Asian Journal of Mathematics},
Volume = {10},
Number = {3},
Pages = {561606},
Year = {2006},
Month = {September},
url = {http://arxiv.org/abs/math/0508428},
Keywords = {almost complex manifolds • quasiintegrable •
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 6manifolds. In particular, I define a 2
parameter family of intrinsic firstorder functionals on
almost complex structures on 6manifolds and compute their
EulerLagrange 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 6sphere admits such nontrivial bundles. This class
of almost complex manifolds in dimension 6 will be referred
to as quasiintegrable. Some of the properties of
quasiintegrable structures (both almost complex and
unitary) are developed and some examples are given. However,
it turns out that quasiintegrability is not an involutive
condition, so the full generality of these structures in
Cartan's sense is not wellunderstood.},
Key = {fds243386}
}
@article{fds318260,
Author = {BRYANT, RL},
Title = {Conformal geometry and 3plane fields on
6manifolds},
Volume = {1502 (Developments of Cartan Geometry an},
Series = {RIMS Symposium Proceedings},
Pages = {115},
Booktitle = {Developments of Cartan Geometry and Related Mathematical
Problems},
Publisher = {Kyoto University},
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
3plane fields D on 6manifolds 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 Hbundle over M where
H is the subgroup of SO(4,3) that stabilizes a null 3plane
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 C^{n+1}
with fixed loci},
Journal = {Chinese Annals of Mathematics, Series B},
Volume = {27},
Number = {1},
Pages = {95112},
Year = {2006},
Month = {January},
MRNUMBER = {MR2209954},
url = {http://arxiv.org/abs/math/0402201},
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 realanalytic arc A (that may be
empty). If n>2, then A has no compact component. Conversely,
an embedded, noncompact nonsingular realanalytic 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.},
Key = {fds243387}
}
@article{fds318261,
Author = {Bryant, R and Freed, D},
Title = {ShiingShen Chern  Obituary},
Journal = {Physics today},
Volume = {59},
Number = {1},
Pages = {7072},
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
3folds},
Journal = {Perspectives in Riemannian Geometry, CRM Proceedings and
Lecture Notes, edited by Vestislav Apostolov, Andrew Dancer,
Nigel Hitchin, and McKenzie Wang, vol. 40 (2006), American
Mathematical Society},
Volume = {40},
Series = {CRM Proceedings and Lecture Notes},
Pages = {6398},
Booktitle = {Perspectives in Riemannian Geometry},
Publisher = {American Mathematical Society},
Editor = {Vestislav Apostolov and Andrew Dancer and Nigel Hitchin and McKenzie Wang},
Year = {2006},
ISBN = {0821838520},
url = {http://arxiv.org/abs/math/0007128},
Abstract = {A second order family of special Lagrangian submanifolds of
complex mspace 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 3space 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 LawlorHarvey
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 = {Contemporary Mathematics},
Volume = {395},
Series = {Contemporary Mathematics},
Pages = {2938},
Booktitle = {150 years of mathematics at Washington University in St.
Louis},
Publisher = {AMS},
Year = {2006},
ISBN = {082183603X},
MRNUMBER = {MR2206889},
url = {http://www.ams.org/mathscinetgetitem?mr=2206889},
Keywords = {53C29 (70F25)},
Key = {fds318263}
}
@article{fds318264,
Author = {Bryant, RL},
Title = {Geodesically reversible Finsler 2spheres of constant
curvature},
Volume = {11},
Series = {Nankai Tracts in Mathematics},
Pages = {95111},
Booktitle = {Inspired by S. S. ChernA Memorial Volume in Honor of a
Great Mathematician},
Publisher = {World Scientific Publishers},
Editor = {Griffiths, PA},
Year = {2006},
Month = {Winter},
url = {http://arxiv.org/abs/math/0407514},
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 2sphere 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 2sphere is necessarily a
Riemannian metric of constant Gauss curvature, thus settling
a longstanding problem in Finsler geometry.},
Key = {fds318264}
}
@article{fds318265,
Author = {Bryant, RL},
Title = {Some remarks on G_2structures},
Pages = {75109},
Booktitle = {Proceedings of Gökova GeometryTopology Conference
2005},
Publisher = {International Press},
Editor = {Akbulut, S and Onder, T and Stern, R},
Year = {2006},
ISBN = {1571461523},
url = {http://arxiv.org/abs/math/0305124},
Abstract = {This article consists of some loosely related remarks about
the geometry of G_2structures on 7manifolds 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_2structure in terms of its torsion. When the fundamental
3form of the G_2structure 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 torsionfree. This version contains some new
results on the pinching of Ricci curvature for metrics
associated to closed G_2structures. 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 longtime existence for given
initial data.},
Key = {fds318265}
}
@article{fds318266,
Author = {Bryant, R},
Title = {Holonomy and Special Geometries},
Series = {Conference Proceedings and Lecture Notes in Geometry and
Topology},
Pages = {7190},
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 = {1571461752},
MRNUMBER = {MR2205367},
url = {http://www.ams.org/mathscinetgetitem?mr=2205367},
Key = {fds318266}
}
@article{fds243379,
Author = {R. Bryant and Bryant, R and Edelsbrunner, H and Koehl, P and Levitt,
M},
Title = {The area derivative of a spacefilling diagram},
Journal = {Discrete and Computanional Geometry},
Volume = {32},
Number = {3},
Pages = {293308},
Year = {2004},
MRNUMBER = {2005k:92077},
url = {http://dx.doi.org/10.1007/s0045400410991},
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
spacefilling 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
inclusionexclusion formulas for these sizes.},
Doi = {10.1007/s0045400410991},
Key = {fds243379}
}
@article{fds10364,
Title = {Leviflat minimal hypersurfaces in twodimensional complex
space forms},
Volume = {37},
Series = {Adv. Stud. Pure Math.},
Pages = {144},
Booktitle = {Lie groups, geometric structures and differential
equationsone 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 Leviflat and minimal. The main results are as follows:
When the curvature of the complex space form is nonzero,
there is a 1parameter family of such hypersurfaces.
Specifically, for each oneparameter subgroup of the
isometry group of the complex space form, there is an
essentially unique example that is invariant under this
oneparameter subgroup. On the other hand, when the
curvature of the space form is zero, i.e., when the space
form is complex 2space 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{fds243380,
Author = {Bryant, RL},
Title = {Some remarks on Finsler manifolds with constant flag
curvature},
Journal = {Houston Journal of Mathematics},
Volume = {28},
Number = {2},
Pages = {221262},
Year = {2002},
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 nmanifold
of constant positive flag curvature out of a hypersurface in
suitably general position in complex projective nspace. The
third remark is that there is a description of the Finsler
metrics of constant curvature on the 2sphere in terms of a
Riemannian metric and 1form on the space of its geodesics.
In particular, this allows one to use any (Riemannian) Zoll
metric of positive Gauss curvature on the 2sphere to
construct a global Finsler metric of constant positive
curvature on the 2sphere. 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
2nmanifolds. As a by product, it is found that these
groups do occur as the holonomy of torsionfree affine
connections in dimension 2n, a hitherto unsuspected
phenomenon. },
Key = {fds243380}
}
@article{MR2002i:53010,
Author = {Bryant, RL},
Title = {On surfaces with prescribed shape operator},
Journal = {Results Math. 40 (2001), no. 14, 88121},
Volume = {40},
Number = {14},
Pages = {88121},
Year = {2001},
MRNUMBER = {2002i:53010},
url = {http://arxiv.org/abs/math/0107083},
Abstract = {The problem of immersing a simply connected surface with a
prescribed shape operator is discussed. From classical and
more recent work, it is known 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
have been classified and it is known that there are two such
families of shape operators, one depending essentially on
three arbitrary functions of one variable (called Type I in
this article) and another depending essentially on two
arbitrary functions of one variable (called Type II in this
article). In this article, these classification results are
rederived, with an emphasis on explicit computability of the
space of solutions. It is shown that, for operators of
either type, their realizations by immersions can be
computed by quadrature. Moreover, explicit normal forms for
each can be computed by quadrature together with, in the
case of Type I, by solving a single linear second order ODE
in one variable. (Even this last step can be avoided in most
Type I cases.) The space of realizations is discussed in
each case, along with some of their remarkable geometric
properties. Several explicit examples are constructed
(mostly already in the literature) and used to illustrate
various features of the problem.},
Key = {MR2002i:53010}
}
@article{fds243382,
Author = {Bryant, RL},
Title = {BochnerKähler metrics},
Journal = {Journal of the AMS},
Volume = {14},
Number = {3},
Pages = {623715},
Year = {2001},
MRNUMBER = {2002i:53096},
url = {http://arxiv.org/abs/math/0003099},
Abstract = {A Kahler metric is said to be BochnerKahler 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 BochnerKahler metrics in
complex dimension n has real dimension n+1 and a recipe for
an explicit formula for any BochnerKahler metric is given.
It is shown that any BochnerKahler metric in complex
dimension n has local (real) cohomogeneity at most~n. The
BochnerKahler 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 BochnerKahler
metric. The fundamental technique is to construct a
canonical infinitesimal torus action on a BochnerKahler
metric whose associated momentum mapping has the orbits of
its symmetry pseudogroupoid as fibers.},
Key = {fds243382}
}
@article{fds243383,
Author = {Bryant, RL},
Title = {Recent advances in the theory of holonomy},
Journal = {Asterisque},
Volume = {266},
Number = {5},
Pages = {351374},
Publisher = {Centre National de la Recherche Scientifique},
Year = {2000},
MRNUMBER = {2001h:53067},
url = {http://www.dmi.ens.fr/bourbaki/Prog_juin99.html},
Key = {fds243383}
}
@article{fds243384,
Author = {Bryant, RL},
Title = {Harmonic morphisms with fibers of dimension
one},
Journal = {Communications in Analysis and Geometry},
Volume = {8},
Number = {2},
Pages = {219265},
Year = {2000},
MRNUMBER = {2001i:53101},
url = {http://arxiv.org/abs/dgga/9701002},
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.},
Key = {fds243384}
}
@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 = {34},
Pages = {405435},
Year = {2000},
MRNUMBER = {2002j:53063},
url = {http://arxiv.org/abs/math/9912246},
Abstract = {Every closed, oriented, real analytic Riemannian 3manifold
can be isometrically embedded as a special Lagrangian
submanifold of a CalabiYau 3fold, even as the real locus
of an antiholomorphic, isometric involution. Every closed,
oriented, real analytic Riemannian 4manifold whose bundle
of selfdual 2forms is trivial can be isometrically
embedded as a coassociative submanifold in a G2manifold,
even as the fixed locus of an antiG2 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.},
Key = {fds243409}
}
@article{fds318269,
Author = {Bryant, R},
Title = {Élie Cartan and geometric duality},
Journal = {Journées Élie Cartan 1998 et 1999},
Volume = {16},
Pages = {520},
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{fds318270,
Author = {Bryant, RL},
Title = {PseudoRiemannian metrics with parallel spinor fields and
vanishing Ricci tensor},
Volume = {4},
Series = {Séminaires & Congrès},
Pages = {5394},
Booktitle = {Global analysis and harmonic analysis (MarseilleLuminy,
1999)},
Publisher = {Société Mathématique de France},
Editor = {Bourguinon, JP and Branson, T and Hijazi, O},
Year = {2000},
ISBN = {2856290949},
MRNUMBER = {2002h:53082},
url = {http://arxiv.org/abs/math/0004073},
Abstract = {I discuss geometry and normal forms for pseudoRiemannian
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 pseudoRiemannian
manifolds, is not an automatic consequence of the existence
of a nontrivial parallel spinor field).},
Key = {fds318270}
}
@article{fds243402,
Author = {Bryant, RL},
Title = {Some examples of special Lagrangian tori},
Journal = {Adv. Theor. Math. Phys.},
Volume = {3},
Number = {1},
Pages = {8390},
Year = {1999},
MRNUMBER = {2000f:32033},
url = {http://arxiv.org/abs/math/9902076},
Abstract = {A short paper giving some examples of smooth hypersurfaces M
of degree n+1 in complex projective nspace that are defined
by real polynomial equations and whose real slice contains a
component diffeomorphic to an n1 torus, which is then
special Lagrangian with respect to the CalabiYau metric on
M.},
Key = {fds243402}
}
@article{fds243408,
Author = {R. Bryant and Sharpe, E and Bryant, RL},
Title = {Dbranes and Spin^cstructures},
Journal = {Physics Letters, Section B: Nuclear, Elementary Particle and
HighEnergy Physics},
Volume = {450},
Number = {4},
Pages = {353357},
Year = {1999},
MRNUMBER = {2000c:53054},
url = {http://arxiv.org/abs/hepth/98l2084},
Abstract = {It was recently pointed out by E. Witten that for a Dbrane
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.},
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 = {183–209},
Year = {1998},
MRNUMBER = {99f:90029},
url = {http://www.ams.org/mathscinetgetitem?mr=99f:90029},
Key = {fds10011}
}
@article{fds243403,
Author = {Bryant, RL},
Title = {Projectively flat Finsler 2spheres of constant
curvature},
Journal = {Selecta Math. (N.S.)},
Volume = {3},
Number = {2},
Pages = {161203},
Year = {1997},
MRNUMBER = {98i:53101},
url = {http://arxiv.org/abs/dgga/9611010},
Key = {fds243403}
}
@article{fds8915,
Title = {Finsler structures on the 2sphere satisfying
K=1},
Volume = {196},
Series = {Contemporary Mathematics},
Pages = {27–41},
Booktitle = {Finsler geometry (Seattle, WA, 1995)},
Publisher = {Amer. Math. Soc., Providence, RI},
Editor = {David Bao and Shiingshen Chern and Zhongmin
Shen},
Year = {1996},
MRNUMBER = {97e:53128},
url = {http://www.math.duke.edu/preprints/9511.dvi},
Key = {fds8915}
}
@article{fds318271,
Author = {Bryant, RL},
Title = {On extremals with prescribed Lagrangian densities},
Volume = {36},
Series = {Symposia Mathematica},
Pages = {86111},
Booktitle = {Manifolds and geometry (Pisa, 1993)},
Publisher = {Cambridge University Press},
Editor = {Bartolomeis, P and Tricerri, F and Vesentini, E},
Year = {1996},
ISBN = {0521562163},
MRNUMBER = {99a:58043},
url = {http://arxiv.org/abs/dgga/9406001},
Abstract = {Consider two manifolds~$M^m$ and $N^n$ and a firstorder
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 EulerLagrange
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 EulerLagrange
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{fds318272,
Author = {Bryant, R},
Title = {Classical, exceptional, and exotic holonomies: a status
report},
Volume = {1},
Series = {Sémin. Congr.},
Pages = {93165},
Booktitle = {Actes de la Table Ronde de Géométrie Différentielle},
Publisher = {Société Mathématique de France},
Editor = {Besse, A},
Year = {1996},
ISBN = {2856290477},
MRNUMBER = {98c:53037},
url = {http://www.math.duke.edu/preprints/9510.dvi},
Abstract = {A survey paper on the status of the holonomy problem as of
1995.},
Key = {fds318272}
}
@article{fds243407,
Author = {R. Bryant and Bryant, RL and Griffiths, PA},
Title = {Characteristic cohomology of differential systems. I.
General theory},
Journal = {The Journal of the American Mathematical
Society},
Volume = {8},
Number = {3},
Pages = {507507},
Year = {1995},
Month = {September},
MRNUMBER = {96c:58183},
url = {http://www.math.duke.edu/preprints/9301.dvi},
Doi = {10.1090/S0894034719951311820X},
Key = {fds243407}
}
@article{fds243404,
Author = {R. Bryant and Griffiths, PA and Hsu, L and Bryant, RL},
Title = {Hyperbolic exterior differential systems and their
conservation laws, Part II},
Journal = {Selecta Math. (N.S.)},
Volume = {1},
Number = {2},
Pages = {265323},
Year = {1995},
MRNUMBER = {97d:580009},
url = {http://www.math.duke.edu/preprints/9413.dvi},
Key = {fds243404}
}
@article{fds243405,
Author = {R. Bryant and Griffiths, PA and Hsu, L and Bryant, RL},
Title = {Hyperbolic exterior differential systems and their
conservation laws, Part I},
Journal = {Selecta Math. (N.S.)},
Volume = {1},
Number = {1},
Pages = {21112},
Year = {1995},
MRNUMBER = {97d:580008},
url = {http://www.math.duke.edu/preprints/9413.dvi},
Key = {fds243405}
}
@article{fds243406,
Author = {R. Bryant and Bryant, RL and Griffiths, PA},
Title = {Characteristic cohomology of differential systems, II:
Conservation laws for a class of parabolic
equations},
Journal = {Duke Math. Journal},
Volume = {78},
Number = {3},
Pages = {531676},
Year = {1995},
MRNUMBER = {96d:58158},
url = {http://www.math.duke.edu/preprints/9302.dvi},
Key = {fds243406}
}
@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 = {176},
Booktitle = {Geometry, Topology, & Physics},
Publisher = {Internat. Press, Cambridge, MA},
Editor = {S.T. Yau},
Year = {1995},
ISBN = {1571460241},
MRNUMBER = {97b:58005},
url = {http://www.math.duke.edu/preprints/9412.dvi},
Key = {fds318273}
}
@article{fds318274,
Author = {R. Bryant and Bryant, R and Gardner, RB},
Title = {Control Structures},
Volume = {12},
Series = {Banach Center Publications},
Pages = {111121},
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/9411.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 = {5181},
Booktitle = {Geometry and quantum field theory (Park City, UT,
1991)},
Publisher = {American Mathematical Society},
Editor = {Freed, D and Uhlenbeck, K},
Year = {1995},
ISBN = {0821804006},
MRNUMBER = {96i:58002},
url = {http://www.ams.org/mathscinetgetitem?mr=96i:58002},
Abstract = {A series of lectures on Lie groups and symplectic geometry,
aimed at the beginning graduate student level.},
Key = {fds318275}
}
@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 = {435461},
Year = {1993},
ISSN = {00209910},
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 = {Bol. Soc. Brasil. Mat. (N.S.)},
Volume = {21},
Number = {2},
Pages = {133157},
Year = {1991},
MRNUMBER = {92k:53112},
url = {http://www.math.duke.edu/preprints/9003.dvi},
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 = {3388},
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 = {0821814923},
MRNUMBER = {93e:53030},
url = {http://www.math.duke.edu/~bryant/ExoticHol.dvi},
Key = {fds318276}
}
@article{fds243398,
Author = {R. Bryant and Harvey, FR and Bryant, RL},
Title = {Submanifolds in hyperKähler geometry},
Journal = {J. Amer. Math. Soc.},
Volume = {2},
Number = {1},
Pages = {131},
Year = {1989},
MRNUMBER = {89m:53090},
url = {http://www.ams.org/mathscinetgetitem?mr=89m:53090},
Key = {fds243398}
}
@article{fds243399,
Author = {R. Bryant and Salamon, S and Bryant, RL},
Title = {On the construction of some complete metrics with
exceptional holonomy},
Journal = {Duke Math. J.},
Volume = {58},
Number = {3},
Pages = {829850},
Year = {1989},
MRNUMBER = {90i:53055},
url = {http://www.ams.org/mathscinetgetitem?mr=90i:53055},
Key = {fds243399}
}
@article{fds318277,
Author = {Bryant, R},
Title = {Surfaces in conformal geometry},
Volume = {48},
Series = {Proc. Sympos. Pure Math.},
Pages = {227240},
Booktitle = {The mathematical heritage of Hermann Weyl (Durham, NC,
1987)},
Publisher = {American Mathematical Society},
Editor = {Wells, RO},
Year = {1988},
ISBN = {0821814826},
MRNUMBER = {89m:53102},
url = {http://www.ams.org/mathscinetgetitem?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 3space.},
Key = {fds318277}
}
@article{fds318278,
Author = {Bryant, R},
Title = {Surfaces of mean curvature one in hyperbolic
space},
Volume = {154155},
Series = {Astérisque},
Pages = {321347},
Booktitle = {Théorie des variétés minimales et applications
(Palaiseau, 1983–1984)},
Publisher = {Société Mathématique de France},
Year = {1988},
MRNUMBER = {955072},
url = {http://www.ams.org/mathscinetgetitem?mr=955072},
Key = {fds318278}
}
@article{fds243397,
Author = {Bryant, RL},
Title = {Metrics with exceptional holonomy},
Journal = {Ann. of Math. (2)},
Volume = {126},
Number = {3},
Pages = {525576},
Year = {1987},
MRNUMBER = {89b:53084},
url = {http://www.ams.org/mathscinetgetitem?mr=89b:53084},
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 = {6576},
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 = {082185075X},
MRNUMBER = {89f:58037},
url = {http://www.ams.org/mathscinetgetitem?mr=89f:58037},
Key = {fds318280}
}
@article{fds318281,
Author = {Bryant, R},
Title = {A survey of Riemannian metrics with special holonomy
groups},
Pages = {505514},
Booktitle = {Proceedings of the International Congress of Mathematicians.
Vol. 1, 2. (Berkeley, Calif., 1986)},
Publisher = {American Mathematical Society},
Editor = {Gleason, A},
Year = {1987},
ISBN = {0821801104},
MRNUMBER = {89f:53068},
url = {http://www.ams.org/mathscinetgetitem?mr=89f:53068},
Key = {fds318281}
}
@article{fds318282,
Author = {Bryant, R},
Title = {Minimal Lagrangian submanifolds of KählerEinstein
manifolds},
Volume = {1255},
Series = {Lecture Notes in Math.},
Pages = {112},
Booktitle = {Differential geometry and differential equations (Shanghai,
1985)},
Publisher = {SpringerVerlag},
Editor = {Gu, C and Berger, M and Bryant, RL},
Year = {1987},
ISBN = {354017849X},
MRNUMBER = {88j:53061},
url = {http://www.ams.org/mathscinetgetitem?mr=88j:53061},
Key = {fds318282}
}
@article{fds243396,
Author = {R. Bryant and Griffiths, PA and Bryant, RL},
Title = {Reduction for constrained variational problems and
$\int{1\over 2}k\sp 2\,ds$},
Journal = {Amer. J. Math.},
Volume = {108},
Number = {3},
Pages = {525570},
Year = {1986},
MRNUMBER = {88a:58044},
url = {http://www.ams.org/mathscinetgetitem?mr=88a:58044},
Key = {fds243396}
}
@article{fds243394,
Author = {Bryant, RL},
Title = {Minimal surfaces of constant curvature in
S^n},
Journal = {Trans. Amer. Math. Soc.},
Volume = {290},
Number = {1},
Pages = {259271},
Year = {1985},
MRNUMBER = {87c:53110},
url = {http://www.ams.org/mathscinetgetitem?mr=87c:53110},
Key = {fds243394}
}
@article{fds243395,
Author = {Bryant, RL},
Title = {Lie groups and twistor spaces},
Journal = {Duke Math. J.},
Volume = {52},
Number = {1},
Pages = {223261},
Year = {1985},
MRNUMBER = {87d:58047},
url = {http://www.ams.org/mathscinetgetitem?mr=87d:58047},
Key = {fds243395}
}
@article{fds318283,
Author = {Bryant, R},
Title = {Metrics with holonomy G2 or Spin(7)},
Volume = {1111},
Series = {Lecture Notes in Math.},
Pages = {269277},
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/mathscinetgetitem?mr=87a:53082},
Key = {fds318283}
}
@article{fds243393,
Author = {Bryant, RL},
Title = {A duality theorem for Willmore surfaces},
Journal = {J. Differential Geom.},
Volume = {20},
Number = {1},
Pages = {2353},
Year = {1984},
MRNUMBER = {86j:58029},
url = {http://www.ams.org/mathscinetgetitem?mr=86j:58029},
Key = {fds243393}
}
@article{fds243391,
Author = {R. Bryant and Griffiths, P and Yang, D},
Title = {Characteristics and existence of isometric
embeddings},
Journal = {Duke Math. J.},
Volume = {50},
Number = {4},
Pages = {893994},
Year = {1983},
MRNUMBER = {85d:53027},
url = {http://www.ams.org/mathscinetgetitem?mr=85d:53027},
Key = {fds243391}
}
@article{fds243392,
Author = {R. Bryant and Berger, E and Griffiths, P},
Title = {The Gauss equations and rigidity of isometric
embeddings},
Journal = {Duke Math. J.},
Volume = {50},
Number = {3},
Pages = {803892},
Year = {1983},
MRNUMBER = {85k:53056},
url = {http://www.ams.org/mathscinetgetitem?mr=85k:53056},
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 = {77102},
Booktitle = {Arithmetic and geometry, Vol. II},
Publisher = {Birkhäuser Boston},
Editor = {Artin, M and Tate, J},
Year = {1983},
ISBN = {376433133X},
MRNUMBER = {86a:32044},
url = {http://www.ams.org/mathscinetgetitem?mr=86a:32044},
Key = {fds318284}
}
@article{fds243389,
Author = {Bryant, RL},
Title = {Holomorphic curves in Lorentzian CRmanifolds},
Journal = {Trans. Amer. Math. Soc.},
Volume = {272},
Number = {1},
Pages = {203221},
Year = {1982},
MRNUMBER = {83i:32029},
url = {http://www.ams.org/mathscinetgetitem?mr=83i:32029},
Abstract = {When can a real hypersurface in complex nspace contain any
complex curves? Since the tangent spaces to such a curve
would have to be null vectors for the Levi form, a necessary
condition is that the Levi form have zeros. The simplest way
this can happen in the nondegenerate case is for the Levi
form to have the Lorentzian signature. In this paper, I show
that a Lorentzian CRmanifold M has at most a finite
parameter family of holomorphic curves, in fact, at most an
n2 parameter family if the dimension of M is 2n+1. This
maximum is attained, as I show by example. When n=2, the
only way it can be reached is for M to be CRflat. In higher
dimensions, where the CRflat model does not achieve the
maximum, it is still unknown whether or not there is more
than one local model with the maximal dimension family of
holomorphic curves. The technique used is exterior
differential systems together with the ChernMoser theory in
the n=2 case. Reprints are available, but can also be
downloaded from the AMS or from JSTOR},
Key = {fds243389}
}
@article{fds243390,
Author = {Bryant, RL},
Title = {Submanifolds and special structures on the
octonians},
Journal = {J. Differential Geom.},
Volume = {17},
Number = {2},
Pages = {185232},
Year = {1982},
MRNUMBER = {84h:53091},
url = {http://www.ams.org/mathscinetgetitem?mr=84h:53091},
Abstract = {A study of the geometry of submanifolds of real 8space
under the group of motions generated by translations and
rotations in the subgroup Spin(7) instead of the full SO(8).
I call real 8space endowed with this group O or octonian
space. The fact that the stabilizer of an oriented 2plane
in Spin(7) is U(3) implies that any oriented 6manifold in O
inherits a U(3)structure. The first part of the paper
studies the generality of the 6manifolds 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 6sphere in O as an
almost complex manifold and study the space of what are now
called pseudoholomorphic curves in the 6sphere. I prove
that every compact Riemann surface occurs as a (possibly
ramified) pseudoholomorphic curve in the 6sphere. I also
show that all of the genus zero pseudoholomorphic curves in
the 6sphere are algebraic as surfaces. Reprints are
available.},
Key = {fds243390}
}
@article{fds243410,
Author = {Bryant, RL},
Title = {Conformal and minimal immersions of compact surfaces into
the 4sphere},
Journal = {J. Differential Geom.},
Volume = {17},
Number = {3},
Pages = {455473},
Year = {1982},
MRNUMBER = {84a:53062},
url = {http://www.ams.org/mathscinetgetitem?mr=84a:53062},
Key = {fds243410}
}
@article{fds318285,
Author = {R. Bryant and Bryant, R and Chern, SS and Griffiths, PA},
Title = {Exterior Differential Systems},
Volume = {1},
Pages = {219338},
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 = {0677164203},
MRNUMBER = {85k:58005},
url = {http://www.ams.org/mathscinetgetitem?mr=85k:58005},
Key = {fds318285}
}
@article{fds243388,
Author = {R. Bryant and Berger, E and Griffiths, P},
Title = {Some isometric embedding and rigidity results for Riemannian
manifolds},
Journal = {Proc. Nat. Acad. Sci. U.S.A.},
Volume = {78},
Number = {8},
Pages = {46574660},
Year = {1981},
MRNUMBER = {82h:53074},
url = {http://www.ams.org/mathscinetgetitem?mr=82h:53074},
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 GelfandGelfand
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{fds320294,
Author = {Bryant, RL},
Title = {On the convex PfaffDarboux Theorem of Ekeland and
Nirenberg},
Year = {2015},
Month = {December},
url = {http://arxiv.org/abs/1512.07100},
Abstract = {The classical PfaffDarboux Theorem, which provides local
`normal forms' for 1forms on manifolds, has applications in
the theory of certain economic models. However, the normal
forms needed in these models come with an additional
requirement of convexity, which is not provided by the
classical proofs of the PfaffDarboux 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 nspace, convexity has
its usual meaning. In 2002, Ekeland and Nirenberg were able
to characterize necessary and sufficient conditions for a
given 1form to admit a convex local normal form (and to
show that some earlier attempts at this characterization had
been unsuccessful). In this article, after providing some
necessary background, I prove a strengthened and generalized
convex PfaffDarboux Theorem, one that covers the case of a
Legendrian foliation in which the notion of convexity is
defined in terms of a torsionfree affine connection on the
underlying manifold. (The main result in Ekeland and
Nirenberg's paper concerns the case in which the affine
connection is flat.)},
Key = {fds320294}
}
@article{fds320295,
Author = {Bryant, RL},
Title = {On the conformal volume of 2tori},
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{fds320296,
Author = {Bryant, RL},
Title = {S.S. Chern's study of almostcomplex structures on the
sixsphere},
Year = {2014},
Month = {May},
url = {http://arxiv.org/abs/1405.3405},
Keywords = {6sphere • complex structure • exceptional
geometry},
Abstract = {In 2003, S.s. Chern began a study of almostcomplex
structures on the 6sphere, with the idea of exploiting the
special properties of its wellknown almostcomplex
structure invariant under the exceptional group $G_2$. While
he did not solve the (currently still open) problem of
determining whether there exists an integrable
almostcomplex structure on the 6sphere, he did prove a
significant identity that resolves the question for an
interesting class of almostcomplex structures on the
6sphere.},
Key = {fds320296}
}
@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 notsotypical applications to illustrate their
use.},
Key = {fds320297}
}
@article{fds225242,
Author = {R.L. Bryant and Feng Xu},
Title = {Laplacian flow for closed G_{2}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}
}
@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 2manifold X
endowed with a holomorphic volume form. A strictly
pseudoconvex real hypersurface M in X inherits not only a
CRstructure 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}
}
