Publications of Robert L Bryant :chronological combined listing:
%% Books
@book{fds25927,
Author = {R.L. Bryant and David Bao and S.-S. Chern and Zhongmin Shen},
Title = {A sampler of Riemann-Finsler geometry},
Volume = {50},
Series = {Mathematical Sciences Research Institute
Publications},
Pages = {xii+363},
Publisher = {Cambridge University Press, Cambridge},
Editor = {Bao, David and Bryant, Robert L. and Chern,
Shiing-Shen},
Year = {2004},
MRNUMBER = {MR2132655(2005j:53003)},
Key = {fds25927}
}
@book{fds10359,
Author = {R.L. Bryant and Phillip Griffiths and Dan Grossmann},
Title = {Exterior Differential Systems and Euler-Lagrange Partial
Differential Equations},
Series = {Chicago Lectures in Mathematics},
Publisher = {University of Chicago Press},
Year = {2003},
MRNUMBER = {MR1985469},
Key = {fds10359}
}
@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},
Key = {fds43013}
}
@book{fds10363,
Title = {Rigidity and quasi-rigidity of extremal cycles in Hermitian
symmetric spaces},
Journal = {Annals of Mathematics Studies (to appear)},
Key = {fds10363}
}
@book{fds10112,
Author = {R.L. Bryant and S.-S. Chern and Robert B. Gardner and Hubert L. Goldschmidt and and
Phillip A. Griffiths},
Title = {Exterior Differential Systems},
Journal = {MSRI Publications18. Springer-Verlag, New York,
1991},
MRNUMBER = {92h:58007},
Key = {fds10112}
}
@book{fds10114,
Author = {R.L. Bryant and Victor Guillemin and Sigurdur Helgason and R. O. Wells, Jr.},
Title = {Integral Geometry},
Journal = {Proceedings of the AMS-IMS-SIAM joint summer research
conference held in Brunswick, Maine, August 12--18, 1984,
pp. x+350, 1987, American Mathematical Society,},
MRNUMBER = {87j:53003},
Key = {fds10114}
}
@book{fds10113,
Author = {R.L. 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},
Key = {fds10113}
}
%% Papers Published
@article{fds146224,
Author = {R.L. Bryant and G. Manno and V. Matveev},
Title = {A solution of a problem of Sophus Lie: Normal forms of
2-dimensional metrics admitting two projective vector
fields},
Journal = {Mathematische Annalen},
Volume = {340},
Number = {2},
Pages = {437--463},
Year = {2008},
Month = {Spring},
url = {http://www.arxiv.org/abs/0705.3592},
Abstract = {We give a complete list of normal forms for the
2-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.},
Key = {fds146224}
}
@article{fds49082,
Title = {Geodesically reversible Finsler 2-spheres of constant
curvature},
Volume = {11},
Series = {Nankai Tracts in Mathematics},
Booktitle = {Inspired by S. S. Chern---A Memorial Volume in Honor of a
Great Mathematician},
Publisher = {World Scientific},
Editor = {P. A. Griffiths},
Year = {2006},
Month = {Winter},
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.},
Key = {fds49082}
}
@article{fds49065,
Title = {On the geometry of almost complex 6-manifolds},
Journal = {The Asian Journal of Mathematics},
Volume = {10},
Number = {3},
Pages = {561--606},
Year = {2006},
Month = {September},
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.},
Key = {fds49065}
}
@article{fds49078,
Title = {Conformal geometry and 3-plane fields on
6-manifolds},
Volume = {1502},
Series = {RIMS Symposium Proceedings},
Pages = {1-15},
Booktitle = {Developments of Cartan Geometry and Related Mathematical
Problems},
Publisher = {Kyoto University},
Year = {2006},
Month = {July},
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 = {fds49078}
}
@article{fds47819,
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 = {95--112},
Year = {2006},
Month = {January},
MRNUMBER = {MR2209954},
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.},
Key = {fds47819}
}
@article{fds49066,
Title = {Second order families of special Lagrangian
3-folds},
Volume = {40},
Series = {CRM Proceedings and Lecture Notes},
Booktitle = {Perspectives in Riemannian Geometry},
Publisher = {American Mathematical Society},
Editor = {Vestislav Apostolov and Andrew Dancer and Nigel Hitchin and McKenzie Wang},
Year = {2006},
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. <p> 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 = {fds49066}
}
@article{fds49067,
Title = {Some remarks on G2 structures},
Booktitle = {Proceeding of Gökova Geometry-Topology Conference
2005},
Publisher = {International Press},
Editor = {S. Akbulut and T Önder and R.J. Stern},
Year = {2006},
Abstract = {This article consists of some loosely related remarks about
the geometry of G_2-structures on 7-manifolds and is 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.<p> 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 = {fds49067}
}
@article{fds47820,
Title = {Geometry of manifolds with special holonomy: "100 years of
holonomy"},
Volume = {395},
Series = {Contemporary Mathematics},
Pages = {29--38},
Booktitle = {150 years of mathematics at Washington University in St.
Louis},
Publisher = {AMS},
Year = {2006},
MRNUMBER = {MR2206889},
Keywords = {53C29 (70F25)},
Key = {fds47820}
}
@article{fds41150,
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 = {Bourguignon, Branson and Chamseddine, Hijazi and Stanton},
Year = {2005},
MRNUMBER = {MR2205367},
Key = {fds41150}
}
@article{fds26204,
Author = {R.L. Bryant and Edelsbrunner, Herbert and Koehl, Patrice and Levitt,
Michael},
Title = {The area derivative of a space-filling diagram},
Journal = {Discrete \& Computational Geometry. An International Journal
of Mathematics and Computer Science},
Volume = {32},
Number = {3},
Pages = {293--308},
Year = {2004},
MRNUMBER = {2005k:92077},
Key = {fds26204}
}
@article{fds10232,
Title = {Some remarks on Finsler manifolds with constant flag
curvature},
Journal = {Houston Journal of Mathematics},
Volume = {28},
Number = {2},
Pages = {221–262},
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. <p> The first remark is that there
is a canonical Kahler structure on the space of geodesics of
such a manifold. <p> 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. <p> 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. <p>
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.<br> },
Key = {fds10232}
}
@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},
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{fds9814,
Title = {Bochner-Kähler metrics},
Journal = {Journal of the AMS},
Volume = {14},
Number = {3},
Pages = {623–715},
Year = {2001},
MRNUMBER = {2002i:53096},
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.},
Key = {fds9814}
}
@article{MR2002i:53010,
Title = {On surfaces with prescribed shape operator},
Journal = {Results in Mathematics},
Volume = {40},
Number = {1–4},
Pages = {88–121},
Year = {2001},
MRNUMBER = {2002i:53010},
Abstract = {The problem of immersing a simply connected surface with a
prescribed shape operator is discussed. <br> I show 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.},
Key = {MR2002i:53010}
}
@article{fds9612,
Title = {Harmonic morphisms with fibers of dimension
one},
Journal = {Communications in Analysis and Geometry},
Volume = {8},
Number = {2},
Pages = {219–265},
Year = {2000},
MRNUMBER = {2001i:53101},
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 = {fds9612}
}
@article{fds9627,
Title = {Recent advances in the theory of holonomy},
Journal = {Astérisque},
Volume = {266},
Number = {5},
Pages = {351–374},
Year = {2000},
MRNUMBER = {2001h:53067},
url = {http://www.dmi.ens.fr/bourbaki/Prog_juin99.html},
Key = {fds9627}
}
@article{fds9628,
Title = {Élie Cartan and geometric duality},
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 = {fds9628}
}
@article{fds9664,
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},
Year = {2000},
MRNUMBER = {2002j:53063},
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.},
Key = {fds9664}
}
@article{fds9815,
Title = {Pseudo-Riemannian metrics with parallel spinor fields and
vanishing Ricci tensor},
Volume = {4},
Series = {Séminaires & Congrès},
Pages = {53–93},
Booktitle = {Global Analysis and Harmonic Analysis},
Year = {2000},
MRNUMBER = {2002h:53082},
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 = {fds9815}
}
@article{fds8906,
Author = {R.L. Bryant and Eric Sharpe},
Title = {D-branes and Spinc-structures},
Journal = {Phys. Lett. B},
Volume = {450},
Number = {4},
Pages = {353–357},
Year = {1999},
MRNUMBER = {2000c:53054},
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.},
Key = {fds8906}
}
@article{fds9210,
Title = {Some examples of special Lagrangian tori},
Journal = {Adv. Theor. Math. Phys.},
Volume = {3},
Number = {1},
Pages = {83–90},
Year = {1999},
MRNUMBER = {2000f:32033},
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.},
Key = {fds9210}
}
@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},
Key = {fds10011}
}
@article{fds8916,
Title = {Projectively flat Finsler 2-spheres of constant
curvature},
Journal = {Selecta Math. (N.S.)},
Volume = {3},
Number = {2},
Pages = {161–203},
Year = {1997},
MRNUMBER = {98i:53101},
Key = {fds8916}
}
@article{fds8913,
Title = {On extremals with prescribed Lagrangian densities},
Volume = {XXXVI},
Series = {Symposia Mathematica},
Pages = {86–111},
Booktitle = {Manifolds and geometry (Pisa, 1993)},
Publisher = {Cambridge Univ. Press, Cambridge},
Editor = {Paolo de Bartolomeis and Franco Tricerri and Edoardo
Vesentini},
Year = {1996},
MRNUMBER = {99a:58043},
Key = {fds8913}
}
@article{fds8914,
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
(Luminy, 1992)},
Publisher = {Soc. Math. France, Paris},
Year = {1996},
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 = {fds8914}
}
@article{fds8915,
Title = {Finsler structures on the 2-sphere 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 Shiing-shen Chern and Zhongmin
Shen},
Year = {1996},
MRNUMBER = {97e:53128},
url = {http://www.math.duke.edu/preprints/95-11.dvi},
Key = {fds8915}
}
@article{fds8907,
Author = {R.L. Bryant and Phillip Griffiths},
Title = {Characteristic cohomology of differential systems, I:
General theory},
Journal = {J. Amer. Math. Soc.},
Volume = {8},
Number = {3},
Pages = {507–596},
Year = {1995},
MRNUMBER = {96c:58183},
url = {http://www.math.duke.edu/preprints/93-01.dvi},
Key = {fds8907}
}
@article{fds8908,
Author = {R.L. Bryant and Phillip Griffiths},
Title = {Characteristic cohomology of differential systems, II:
Conservation laws for a class of parabolic
equations},
Journal = {Duke Math. Journal},
Volume = {78},
Number = {3},
Pages = {531–676},
Year = {1995},
MRNUMBER = {96d:58158},
url = {http://www.math.duke.edu/preprints/93-02.dvi},
Key = {fds8908}
}
@article{fds8909,
Author = {R.L. Bryant and Phillip A. Griffiths and Lucas Hsu},
Title = {Hyperbolic exterior differential systems and their
conservation laws, Part I},
Journal = {Selecta Math. (N.S.)},
Volume = {1},
Number = {1},
Pages = {21–112},
Year = {1995},
MRNUMBER = {97d:580008},
url = {http://www.math.duke.edu/preprints/94-13.dvi},
Key = {fds8909}
}
@article{fds8910,
Author = {R.L. Bryant and Phillip A. Griffiths and Lucas Hsu},
Title = {Hyperbolic exterior differential systems and their
conservation laws, Part II},
Journal = {Selecta Math. (N.S.)},
Volume = {1},
Number = {2},
Pages = {265–323},
Year = {1995},
MRNUMBER = {97d:580009},
url = {http://www.math.duke.edu/preprints/94-13.dvi},
Key = {fds8910}
}
@article{fds8911,
Author = {R.L. Bryant and Phillip Griffiths and Lucas Hsu},
Title = {Toward a geometry of differential equations},
Volume = {IV},
Series = {Conf. Proc. Lecture Notes Geom. Topology},
Pages = {1–76},
Booktitle = {Geometry, Topology, & Physics},
Publisher = {Internat. Press, Cambridge, MA},
Editor = {S.-T. Yau},
Year = {1995},
MRNUMBER = {97b:58005},
url = {http://www.math.duke.edu/preprints/94-12.dvi},
Key = {fds8911}
}
@article{fds8912,
Author = {R.L. Bryant and Robert B. Gardner},
Title = {Control structures},
Volume = {12},
Series = {Banach Center Publications},
Pages = {111–121},
Booktitle = {Geometry in nonlinear control and differential inclusions
(Warsaw, 1993)},
Publisher = {Polish Acad. Sci., Warsaw},
Editor = {Bronis\l aw Jakubczyk and Witold Respondek and Tadeusz
Rze\.zuchowski},
Year = {1995},
MRNUMBER = {96h:93024},
url = {http://www.math.duke.edu/preprints/94-11.dvi},
Key = {fds8912}
}
@article{fds9565,
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 = {Amer. Math. Soc., Providence, RI},
Year = {1995},
MRNUMBER = {96i:58002},
Abstract = {A series of lectures on Lie groups and symplectic geometry,
aimed at the beginning graduate student level.},
Key = {fds9565}
}
@article{fds9566,
Author = {R.L. Bryant and Lucas Hsu},
Title = {Rigidity of integral curves of rank 2 distributions},
Journal = {Invent. Math.},
Volume = {114},
Number = {2},
Pages = {435–461},
Year = {1993},
MRNUMBER = {94j:58003},
url = {http://www.math.duke.edu/~bryant/Rigid.dvi},
Key = {fds9566}
}
@article{fds9567,
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 = {Amer. Math. Soc., Providence, RI},
Editor = {James A. Carlson and C. Herbert Clemens and David R.
Morrison},
Year = {1991},
MRNUMBER = {93e:53030},
url = {http://www.math.duke.edu/~bryant/ExoticHol.dvi},
Key = {fds9567}
}
@article{fds9568,
Title = {Some remarks on the geometry of austere manifolds},
Journal = {Bol. Soc. Brasil. Mat. (N.S.)},
Volume = {21},
Number = {2},
Pages = {133–157},
Year = {1991},
MRNUMBER = {92k:53112},
url = {http://www.math.duke.edu/preprints/90-03.dvi},
Key = {fds9568}
}
@article{fds9570,
Author = {R.L. Bryant and Simon Salamon},
Title = {On the construction of some complete metrics with
exceptional holonomy},
Journal = {Duke Math. J.},
Volume = {58},
Number = {3},
Pages = {829–850},
Year = {1989},
MRNUMBER = {90i:53055},
Key = {fds9570}
}
@article{fds9572,
Author = {R.L. Bryant and F. Reese Harvey},
Title = {Submanifolds in hyper-Kähler geometry},
Journal = {J. Amer. Math. Soc.},
Volume = {2},
Number = {1},
Pages = {1–31},
Year = {1989},
MRNUMBER = {89m:53090},
Key = {fds9572}
}
@article{fds9571,
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 = {Amer. Math. Soc., Providence, RI},
Year = {1988},
MRNUMBER = {89m:53102},
Key = {fds9571}
}
@article{fds9588,
Title = {Surfaces of mean curvature one in hyperbolic
space},
Volume = {154-155 (1987)},
Series = {Astérisque},
Pages = {321–347},
Booktitle = {Théorie des variétés minimales et applications
(Palaiseau, 1983–1984)},
Publisher = {Société Mathématique de France, Paris},
Year = {1988},
MRNUMBER = {955072},
Key = {fds9588}
}
@article{fds9573,
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 = {Amer. Math. Soc., Providence, RI},
Editor = {Mladen Luksic and Clyde Martin and William Shadwick},
Year = {1987},
MRNUMBER = {89f:58037},
Key = {fds9573}
}
@article{fds9574,
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 = {Amer. Math. Soc., Providence, RI},
Year = {1987},
MRNUMBER = {89f:53068},
Key = {fds9574}
}
@article{fds9575,
Title = {Metrics with exceptional holonomy},
Journal = {Ann. of Math. (2)},
Volume = {126},
Number = {3},
Pages = {525–576},
Year = {1987},
MRNUMBER = {89b:53084},
Key = {fds9575}
}
@article{fds10012,
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, Berlin},
Editor = {Chao Hao Gu and Marcel Berger and Robert L.
Bryant},
Year = {1987},
MRNUMBER = {88j:53061},
Key = {fds10012}
}
@article{fds9576,
Author = {R.L. Bryant and Phillip Griffiths},
Title = {Reduction for constrained variational problems and
$\int{1\over 2}k\sp 2\,ds$},
Journal = {Amer. J. Math.},
Volume = {108},
Number = {3},
Pages = {525–570},
Year = {1986},
MRNUMBER = {88a:58044},
Key = {fds9576}
}
@article{fds9577,
Title = {Lie groups and twistor spaces},
Journal = {Duke Math. J.},
Volume = {52},
Number = {1},
Pages = {223–261},
Year = {1985},
MRNUMBER = {87d:58047},
Key = {fds9577}
}
@article{fds9578,
Title = {Minimal surfaces of constant curvature in
Sn},
Journal = {Trans. Amer. Math. Soc.},
Volume = {290},
Number = {1},
Pages = {259–271},
Year = {1985},
MRNUMBER = {87c:53110},
Key = {fds9578}
}
@article{fds9579,
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, Berlin--New York},
Editor = {F. Hirzebruch and J. Schwermer and S. Suter},
Year = {1985},
MRNUMBER = {87a:53082},
Key = {fds9579}
}
@article{fds9580,
Title = {A duality theorem for Willmore surfaces},
Journal = {J. Differential Geom.},
Volume = {20},
Number = {1},
Pages = {23–53},
Year = {1984},
MRNUMBER = {86j:58029},
Key = {fds9580}
}
@article{fds9581,
Author = {R.L. Bryant and Phillip A. Griffiths},
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, Boston, MA},
Editor = {Michael Artin and John Tate},
Year = {1983},
MRNUMBER = {86a:32044},
Key = {fds9581}
}
@article{fds9582,
Author = {R.L. Bryant and Eric Berger and Phillip Griffiths},
Title = {The Gauss equations and rigidity of isometric
embeddings},
Journal = {Duke Math. J.},
Volume = {50},
Number = {3},
Pages = {803–892},
Year = {1983},
MRNUMBER = {85k:53056},
Key = {fds9582}
}
@article{fds9583,
Author = {R.L. Bryant and Phillip Griffiths and Deane Yang},
Title = {Characteristics and existence of isometric
embeddings},
Journal = {Duke Math. J.},
Volume = {50},
Number = {4},
Pages = {893–994},
Year = {1983},
MRNUMBER = {85d:53027},
Key = {fds9583}
}
@article{fds9584,
Title = {Submanifolds and special structures on the
octonians},
Journal = {J. Differential Geom.},
Volume = {17},
Number = {2},
Pages = {185–232},
Year = {1982},
MRNUMBER = {84h:53091},
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 <b>O</b> or
octonian space. <p>The fact that the stabilizer of an
oriented 2-plane in Spin(7) is U(3) implies that any
oriented 6-manifold in <b>O</b> 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. <p>I then turn to the study
of the standard 6-sphere in <b>O</b> 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. <p>Reprints are
available.},
Key = {fds9584}
}
@article{fds9585,
Title = {Conformal and minimal immersions of compact surfaces into
the 4-sphere},
Journal = {J. Differential Geom.},
Volume = {17},
Number = {3},
Pages = {455-473},
Year = {1982},
MRNUMBER = {84a:53062},
Key = {fds9585}
}
@article{fds9586,
Title = {Holomorphic curves in Lorentzian CR-manifolds},
Journal = {Trans. Amer. Math. Soc.},
Volume = {272},
Number = {1},
Pages = {203–221},
Year = {1982},
MRNUMBER = {83i:32029},
Abstract = {When can a real hypersurface in complex n-space 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 non-degenerate case is for the Levi
form to have the Lorentzian signature. <p>In this paper, I
show that a Lorentzian CR-manifold M has at most a finite
parameter family of holomorphic curves, in fact, at most an
n<sup>2</sup> 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 CR-flat.
In higher dimensions, where the CR-flat 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. <p>The technique
used is exterior differential systems together with the
Chern-Moser theory in the n=2 case. <p>Reprints are
available, but can also be downloaded from the AMS or from
JSTOR},
Key = {fds9586}
}
@article{fds10111,
Author = {R.L. Bryant and Shiing-Shen Chern and Phillip Griffiths},
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, Beijing},
Editor = {S. S. Chern and Wen Tsün Wu},
Year = {1982},
MRNUMBER = {85k:58005},
Key = {fds10111}
}
@article{fds9587,
Author = {R.L. Bryant and Eric Berger and Phillip Griffiths},
Title = {Some isometric embedding and rigidity results for Riemannian
manifolds},
Journal = {Proc. Nat. Acad. Sci. U.S.A.},
Volume = {78},
Number = {8},
Pages = {4657–4660},
Year = {1981},
MRNUMBER = {82h:53074},
Key = {fds9587}
}
%% Papers Submitted
@article{fds146226,
Title = {Nonembedding and nonextension results in special
holonomy},
Booktitle = {Proceedings of the August 2006 Madrid conference in honor of
Nigel Hitchin's 60th Birthday},
Publisher = {Oxford University Press},
Editor = {Jean-Pierre Bourguignon and Simon Salamon and Oscar Garcia
Prada},
Year = {2007},
Month = {Fall},
Key = {fds146226}
}
@article{fds146225,
Title = {Gradient Kähler Ricci Solitons},
Booktitle = {Proceedings of the conference `Géométrie différentielle,
Physique mathématique, Mathématiques et
Société'},
Year = {2004},
Month = {July},
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 Poincare coordinates in the case that
the Ricci curvature is positive and the vector field has a
fixed point.},
Key = {fds146225}
}
%% Preprints
@article{fds141021,
Author = {R.L. Bryant and M. Dunajski and M. Eastwood},
Title = {Metrisability of two-dimensional projective
structures},
Year = {2008},
Abstract = {We carry out the programme of R. Liouville \cite{Liouville}
to construct an explicit local obstruction to the existence
of a Levi--Civita 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 = {fds141021}
}
@article{fds24759,
Title = {Real hypersurfaces in unimodular complex
surfaces},
Year = {2004},
Month = {July},
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 = {fds24759}
}
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Mathematics Department