%% 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}, 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 Riemann-Finsler Geometry}, Volume = {50}, Series = {Mathematical Sciences Research Institute Publications}, Pages = {363 pages}, Publisher = {Cambridge University Press}, Editor = {Bao, D and Bryant, RL and Chern, S-S and Shen, Z}, Year = {2004}, Month = {November}, ISBN = {0521831814}, MRNUMBER = {MR2132655(2005j:53003)}, 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, R and Griffiths, P and Grossman, D}, Title = {Exterior Differential Systems and Euler-Lagrange Partial Differential Equations}, Series = {Chicago Lectures in Mathematics}, Pages = {213 pages}, Publisher = {University of Chicago Press}, Year = {2003}, Month = {July}, ISBN = {0226077934}, MRNUMBER = {MR1985469}, Abstract = {The book also covers the Second Variation, Euler-Lagrange PDE systems, and higher-order conservation laws.}, Key = {fds318268} } @book{fds43013, Title = {Selected works of Phillip A. Griffiths with commentary. Part 4. Differential systems.}, Publisher = {American Mathematical Society, Providence, RI; International Press, Somerville, MA}, Editor = {R. L. Bryant and David R. Morrison}, Year = {2003}, MRNUMBER = {2005e:01025d}, Key = {fds43013} } @book{fds320301, Author = {Bryant, RL}, Title = {Rigidity and quasi-rigidity of extremal cycles in Hermitian symmetric spaces}, Year = {2001}, Month = {March}, Abstract = {I use local differential geometric techniques to prove that the algebraic cycles in certain extremal homology classes in Hermitian symmetric spaces are either rigid (i.e., deformable only by ambient motions) or quasi-rigid (roughly speaking, foliated by rigid subvarieties in a nontrivial way). These rigidity results have a number of applications: First, they prove that many subvarieties in Grassmannians and other Hermitian symmetric spaces cannot be smoothed (i.e., are not homologous to a smooth subvariety). Second, they provide characterizations of holomorphic bundles over compact Kahler manifolds that are generated by their global sections but that have certain polynomials in their Chern classes vanish (for example, c_2 = 0, c_1c_2 - c_3 = 0, c_3 = 0, etc.).}, Key = {fds320301} } @book{fds318279, Author = {R. Bryant and Victor Guillemin and Sigurdur Helgason and R. O. Wells, Jr.}, Title = {Integral Geometry}, Volume = {63}, Pages = {350 pages}, Publisher = {American Mathematical Society}, Editor = {Bryant, R and Guillemin, V and Helgason, S and Wells, RO}, Year = {1987}, ISBN = {0-8218-5071-7}, MRNUMBER = {87j:53003}, Abstract = {Proceedings of the AMS-IMS-SIAM joint summer research conference held in Brunswick, Maine, August 12–18, 1984}, Key = {fds318279} } @book{fds10113, Author = {R. Bryant and Marcel Berger and Chao Hao Gu}, Title = {Differential Geometry and Differential Equations}, Journal = {Proceedings of the sixth symposium held at Fudan University, Shanghai, June 21--July 6, 1985, pp. xii+243, 1987, Springer-Verlag, Berlin}, MRNUMBER = {88b:53002}, 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 = {345-357}, Year = {2017}, Month = {June}, 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 Bernstein-Gelfand-Gelfand complex associated with the trivial representation. However, at least in the cases we discuss, our constructions are relatively simple and avoid most of the machinery of parabolic geometry. Moreover, our method extends to certain geometries beyond the parabolic realm.}, Key = {fds320298} } @article{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 = {346-367}, Booktitle = {The Many Facets of Geometry: A Tribute to Nigel Hitchin}, Publisher = {Oxford University Press}, Address = {Oxford}, Editor = {Garcia-Prada, O and Bourguignon, JP and Salamon, S}, Year = {2010}, Month = {Fall}, ISBN = {0199534926}, MRCLASS = {53C29}, MRNUMBER = {MR2681703}, 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 well-posed 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 = {177-178}, Year = {2009}, ISSN = {0273-0979}, Doi = {10.1090/S0273-0979-09-01248-8}, Key = {fds243377} } @article{fds243378, Author = {R. Bryant and Bryant, RL and Dunajski, M and Eastwood, M}, Title = {Metrisability of two-dimensional projective structures}, Volume = {83}, Number = {3}, Pages = {465-499}, Year = {2009}, ISSN = {0022-040X}, MRCLASS = {53}, MRNUMBER = {MR2581355}, 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 = {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 two-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 two-dimensional metrics that admit a transitive Lie pseudogroup of geodesic-preserving transformations and we show that these normal forms are mutually non-isometric. This solves a problem posed by Sophus Lie. © 2007 Springer-Verlag.}, Doi = {10.1007/s00208-007-0158-3}, Key = {fds243385} } @article{fds320198, Author = {Bryant, RL}, Title = {Gradient Kähler Ricci solitons}, Journal = {Astérisque}, Volume = {321}, Series = {Astérisque}, Number = {321}, Pages = {51-97}, Booktitle = {Géométrie différentielle, physique mathématique, mathématiques et société. I.}, Publisher = {Soc. Math. France}, Year = {2008}, Month = {Spring}, ISBN = {978-285629-258-7}, MRCLASS = {53C55 (53C21)}, MRNUMBER = {2010i:53138}, 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 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 = {fds243386} } @article{fds318260, Author = {BRYANT, RL}, Title = {Conformal geometry and 3-plane fields on 6-manifolds}, Volume = {1502 (Developments of Cartan Geometry an}, 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 = {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 = {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 = {fds243387} } @article{fds318261, Author = {Bryant, R and Freed, D}, Title = {Shiing-Shen Chern - Obituary}, Journal = {Physics Today}, Volume = {59}, Number = {1}, Pages = {70-72}, Year = {2006}, Month = {January}, Doi = {10.1063/1.2180187}, Key = {fds318261} } @article{fds318262, Author = {Bryant, RL}, Title = {Second order families of special Lagrangian 3-folds}, Journal = {Perspectives in Riemannian Geometry}, Volume = {40}, Series = {CRM Proceedings and Lecture Notes}, Pages = {63-98}, Booktitle = {Perspectives in Riemannian Geometry}, Publisher = {American Mathematical Society}, Editor = {Vestislav Apostolov and Andrew Dancer and Nigel Hitchin and McKenzie Wang}, Year = {2006}, ISBN = {0-8218-3852-0}, Abstract = {A second order family of special Lagrangian submanifolds of complex m-space is a family characterized by the satisfaction of a set of pointwise conditions on the second fundamental form. For example, the set of ruled special Lagrangian submanifolds of complex 3-space is characterized by a single algebraic equation on the second fundamental form. While the `generic' set of such conditions turns out to be incompatible, i.e., there are no special Lagrangian submanifolds that satisfy them, there are many interesting sets of conditions for which the corresponding family is unexpectedly large. In some cases, these geometrically defined families can be described explicitly, leading to new examples of special Lagrangian submanifolds. In other cases, these conditions characterize already known families in a new way. For example, the examples of Lawlor-Harvey constructed for the solution of the angle conjecture and recently generalized by Joyce turn out to be a natural and easily described second order family.}, Key = {fds318262} } @article{fds318263, Author = {Bryant, RL}, Title = {Geometry of manifolds with special holonomy: "100 years of holonomy"}, Journal = {Surveys on Discrete and Computational Geometry: Twenty Years Later}, Volume = {395}, Series = {Contemporary Mathematics}, Pages = {29-38}, Booktitle = {150 years of mathematics at Washington University in St. Louis}, Publisher = {AMS}, Year = {2006}, ISBN = {0-8218-3603-X}, MRNUMBER = {MR2206889}, Keywords = {53C29 (70F25)}, Key = {fds318263} } @article{fds318264, Author = {Bryant, R}, Title = {Geodesically reversible Finsler 2-spheres of constant curvature}, Volume = {11}, Series = {Nankai Tracts in Mathematics}, Pages = {95-111}, Booktitle = {Inspired by S. S. Chern---A Memorial Volume in Honor of a Great Mathematician}, Publisher = {World Scientific Publishers}, Editor = {Griffiths, PA}, 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 = {fds318264} } @article{fds318265, Author = {Bryant, RL}, Title = {Some remarks on G_2-structures}, Pages = {75-109}, Booktitle = {Proceedings of Gökova Geometry-Topology Conference 2005}, Publisher = {International Press}, Editor = {Akbulut, S and Onder, T and Stern, R}, Year = {2006}, ISBN = {1-57146-152-3}, Abstract = {This article consists of some loosely related remarks about the geometry of G_2-structures on 7-manifolds and is partly based on old unpublished joint work with two other people: F. Reese Harvey and Steven Altschuler. Much of this work has since been subsumed in the work of Hitchin \cite{MR02m:53070} and Joyce \cite{MR01k:53093}. I am making it available now mainly because of interest expressed by others in seeing these results written up since they do not seem to have all made it into the literature. A formula is derived for the scalar curvature and Ricci curvature of a G_2-structure in terms of its torsion. When the fundamental 3-form of the G_2-structure is closed, this formula implies, in particular, that the scalar curvature of the underlying metric is nonpositive and vanishes if and only if the structure is torsion-free. This version contains some new results on the pinching of Ricci curvature for metrics associated to closed G_2-structures. Some formulae are derived for closed solutions of the Laplacian flow that specify how various related quantities, such as the torsion and the metric, evolve with the flow. These may be useful in studying convergence or long-time existence for given initial data.}, Key = {fds318265} } @article{fds318266, Author = {Bryant, R}, Title = {Holonomy and Special Geometries}, Series = {Conference Proceedings and Lecture Notes in Geometry and Topology}, Pages = {71-90}, Booktitle = {Dirac Operators: Yesterday and Today}, Publisher = {International Press}, Editor = {Bourguinon, JP and Branson, T and Chamseddine, A and Hijazi, O and Stanton, R}, Year = {2005}, ISBN = {1-57146-175-2}, MRNUMBER = {MR2205367}, 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 space-filling diagram}, Journal = {Discrete and Computanional Geometry}, Volume = {32}, Number = {3}, Pages = {293-308}, Year = {2004}, MRNUMBER = {2005k:92077}, Abstract = {The motion of a biomolecule greatly depends on the engulfing solution, which is mostly water. Instead of representing individual water molecules, it is desirable to develop implicit solvent models that nevertheless accurately represent the contribution of the solvent interaction to the motion. In such models, hydrophobicity is expressed as a weighted sum of atomic surface areas. The derivatives of these weighted areas contribute to the force that drives the motion. In this paper we give formulas for the weighted and unweighted area derivatives of a molecule modeled as a space-filling diagram made up of balls in motion. Other than the radii and the centers of the balls, the formulas are given in terms of the sizes of circular arcs of the boundary and edges of the power diagram. We also give inclusion-exclusion formulas for these sizes.}, Doi = {10.1007/s00454-004-1099-1}, Key = {fds243379} } @article{fds243380, Author = {Bryant, RL}, Title = {Some remarks on Finsler manifolds with constant flag curvature}, Journal = {Houston Journal of Mathematics}, Volume = {28}, Number = {2}, Pages = {221-262}, Publisher = {UNIV HOUSTON}, Year = {2002}, Month = {January}, MRNUMBER = {2003h:53102}, url = {HJM}, Abstract = {This article is an exposition of four loosely related remarks on the geometry of Finsler manifolds with constant positive flag curvature. The first remark is that there is a canonical Kahler structure on the space of geodesics of such a manifold. The second remark is that there is a natural way to construct a (not necessarily complete) Finsler n-manifold of constant positive flag curvature out of a hypersurface in suitably general position in complex projective n-space. The third remark is that there is a description of the Finsler metrics of constant curvature on the 2-sphere in terms of a Riemannian metric and 1-form on the space of its geodesics. In particular, this allows one to use any (Riemannian) Zoll metric of positive Gauss curvature on the 2-sphere to construct a global Finsler metric of constant positive curvature on the 2-sphere. The fourth remark concerns the generality of the space of (local) Finsler metrics of constant positive flag curvature in dimension n+1>2 . It is shown that such metrics depend on n(n+1) arbitrary functions of n+1 variables and that such metrics naturally correspond to certain torsion- free S^1 x GL(n,R)-structures on 2n-manifolds. As a by- product, it is found that these groups do occur as the holonomy of torsion-free affine connections in dimension 2n, a hitherto unsuspected phenomenon. }, Key = {fds243380} } @article{fds10364, Title = {Levi-flat minimal hypersurfaces in two-dimensional complex space forms}, Volume = {37}, Series = {Adv. Stud. Pure Math.}, Pages = {1--44}, Booktitle = {Lie groups, geometric structures and differential equations---one hundred years after Sophus Lie (Kyoto/Nara, 1999)}, Publisher = {Math. Soc. Japan}, Year = {2002}, MRNUMBER = {MR1980895}, 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{fds243382, Author = {Bryant, RL}, Title = {Bochner-kähler metrics}, Journal = {Journal of the American Mathematical Society}, Volume = {14}, Number = {3}, Pages = {623-715}, Year = {2001}, Month = {July}, 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.}, Doi = {10.1090/S0894-0347-01-00366-6}, Key = {fds243382} } @article{MR2002i:53010, Author = {Bryant, RL}, Title = {On surfaces with prescribed shape operator}, Journal = {Results Math. 40 (2001), no. 1-4, 88--121}, 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. 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{fds243383, Author = {Bryant, RL}, Title = {Recent advances in the theory of holonomy}, Journal = {Astérisque}, Volume = {266}, Number = {5}, Pages = {351-374}, 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 = {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 = {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 = {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 G2-manifold, even as the fixed locus of an anti-G2 involution. These results, when coupled with McLean's analysis of the moduli spaces of such calibrated sub-manifolds, 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 = {5-20}, Booktitle = {Journées Élie Cartan 1998 et 1999}, Publisher = {Institut Élie Cartan}, Year = {2000}, url = {http://www.math.duke.edu/~bryant/Cartan.pdf}, Key = {fds318269} } @article{fds318270, Author = {Bryant, R}, Title = {Pseudo-Reimannian metrics with parallel spinor fields and vanishing Ricci tensor}, Volume = {4}, Series = {Séminaires & Congrès}, Pages = {53-94}, Booktitle = {Global analysis and harmonic analysis (Marseille-Luminy, 1999)}, Publisher = {Société Mathématique de France}, Editor = {Bourguinon, JP and Branson, T and Hijazi, O}, Year = {2000}, ISBN = {2-85629-094-9}, 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 = {fds318270} } @article{fds243402, Author = {Bryant, RL}, 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 = {fds243402} } @article{fds243408, Author = {R. Bryant and Sharpe, E and Bryant, RL}, Title = {D-branes and Spin^c-structures}, Journal = {Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics}, Volume = {450}, Number = {4}, Pages = {353-357}, 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 = {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}, Key = {fds10011} } @article{fds243403, Author = {Bryant, RL}, Title = {Projectively flat Finsler 2-spheres of constant curvature}, Journal = {Selecta Mathematica}, Volume = {3}, Number = {2}, Pages = {161-203}, Year = {1997}, Month = {March}, MRNUMBER = {98i:53101}, Doi = {10.1007/s000290050009}, Key = {fds243403} } @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{fds318271, Author = {Bryant, R}, Title = {On extremals with prescribed Lagrangian densities}, Volume = {36}, Series = {Symposia Mathematica}, Pages = {86-111}, Booktitle = {Manifolds and geometry (Pisa, 1993)}, Publisher = {Cambridge University Press}, Editor = {Bartolomeis, P and Tricerri, F and Vesentini, E}, Year = {1996}, ISBN = {0-521-56216-3}, MRNUMBER = {99a:58043}, Abstract = {This article studies some examples of the family of problems where a Lagrangian is given for maps from one manifold to another and one is interested in the extremal mappings for which the Lagrangian density takes a prescribed form. The first problem is the study of when two minimal graphs can induce the same area function on the domain without differing by trivial symmetries. The second problem is similar but concerns a different `area Lagrangian' first investigated by Calabi. The third problem classified the harmonic maps between spheres (more generally, manifolds of constant sectional curvature) for which the energy density is a constant multiple of the volume form. In the first and third cases, the complete solution is described. In the second case, some information about the solutions is derived, but the problem is not completely solved.}, Key = {fds318271} } @article{fds318272, Author = {Bryant, R}, Title = {Classical, exceptional, and exotic holonomies: a status report}, Volume = {1}, Series = {Sémin. Congr.}, Pages = {93-165}, Booktitle = {Actes de la Table Ronde de Géométrie Différentielle}, Publisher = {Société Mathématique de France}, Editor = {Besse, A}, Year = {1996}, ISBN = {2-85629-047-7}, MRNUMBER = {98c:53037}, url = {http://www.math.duke.edu/preprints/95-10.dvi}, Abstract = {A survey paper on the status of the holonomy problem as of 1995.}, Key = {fds318272} } @article{fds243407, Author = {R. Bryant and Bryant, RL and Griffiths, PA}, Title = {Characteristic cohomology of differential systems. I. General theory}, Journal = {Journal of the American Mathematical Society}, Volume = {8}, Number = {3}, Pages = {507-507}, Year = {1995}, Month = {September}, MRNUMBER = {96c:58183}, url = {http://www.math.duke.edu/preprints/93-01.dvi}, Doi = {10.1090/S0894-0347-1995-1311820-X}, 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 = {265-323}, Year = {1995}, MRNUMBER = {97d:580009}, url = {http://www.math.duke.edu/preprints/94-13.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 = {21-112}, Year = {1995}, MRNUMBER = {97d:580008}, url = {http://www.math.duke.edu/preprints/94-13.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 = {531-676}, Year = {1995}, MRNUMBER = {96d:58158}, url = {http://www.math.duke.edu/preprints/93-02.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 = {1-76}, Booktitle = {Geometry, Topology, & Physics}, Publisher = {Internat. Press, Cambridge, MA}, Editor = {S.-T. Yau}, Year = {1995}, ISBN = {1-57146-024-1}, MRNUMBER = {97b:58005}, url = {http://www.math.duke.edu/preprints/94-12.dvi}, Key = {fds318273} } @article{fds318274, Author = {R. Bryant and Bryant, R and Gardner, RB}, Title = {Control Structures}, Volume = {12}, Series = {Banach Center Publications}, Pages = {111-121}, Booktitle = {Geometry in nonlinear control and differential inclusions (Warsaw, 1993)}, Publisher = {Polish Academy of Sciences}, Editor = {Jakubczyk, B and Respondek, W and Rzezuchowski, T}, Year = {1995}, MRNUMBER = {96h:93024}, url = {http://www.math.duke.edu/preprints/94-11.dvi}, Key = {fds318274} } @article{fds318275, Author = {Bryant, R}, Title = {An introduction to Lie groups and symplectic geometry}, Volume = {1}, Series = {IAS/Park City Mathematics}, Pages = {5-181}, Booktitle = {Geometry and quantum field theory (Park City, UT, 1991)}, Publisher = {American Mathematical Society}, Editor = {Freed, D and Uhlenbeck, K}, Year = {1995}, ISBN = {0-8218-0400-6}, MRNUMBER = {96i:58002}, 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 = {435-461}, Year = {1993}, Month = {December}, ISSN = {0020-9910}, MRNUMBER = {94j:58003}, url = {http://www.math.duke.edu/~bryant/Rigid.dvi}, Doi = {10.1007/BF01232676}, Key = {fds243401} } @article{fds243400, Author = {Bryant, RL}, Title = {Some remarks on the geometry of austere manifolds}, Journal = {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 = {fds243400} } @article{fds318276, Author = {Bryant, R}, Title = {Two exotic holonomies in dimension four, path geometries, and twistor theory}, Volume = {53}, Series = {Proc. Sympos. Pure Math.}, Pages = {33-88}, Booktitle = {Complex geometry and Lie theory (Sundance, UT, 1989)}, Publisher = {American Mathematical Society}, Editor = {Carlson, J and Clemens, H and Morrison, D}, Year = {1991}, ISBN = {0-8218-1492-3}, MRNUMBER = {93e:53030}, url = {http://www.math.duke.edu/~bryant/ExoticHol.dvi}, Key = {fds318276} } @article{fds243398, Author = {R. Bryant and Harvey, FR and Bryant, RL}, Title = {Submanifolds in hyper-Kähler geometry}, Journal = {J. Amer. Math. Soc.}, Volume = {2}, Number = {1}, Pages = {1-31}, Year = {1989}, MRNUMBER = {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 = {829-850}, Year = {1989}, MRNUMBER = {90i:53055}, Key = {fds243399} } @article{fds318277, Author = {Bryant, R}, Title = {Surfaces in conformal geometry}, Volume = {48}, Series = {Proc. Sympos. Pure Math.}, Pages = {227-240}, Booktitle = {The mathematical heritage of Hermann Weyl (Durham, NC, 1987)}, Publisher = {American Mathematical Society}, Editor = {Wells, RO}, Year = {1988}, ISBN = {0-8218-1482-6}, MRNUMBER = {89m:53102}, Abstract = {A survey paper. However, there are some new results. Building on the results in A duality theorm for Willmore surfaces, I use the Klein correspondance to determine the moduli space of Willmore critical spheres for low critical values and also determine the moduli space of Willmore minima for the real projective plane in 3-space.}, Key = {fds318277} } @article{fds318278, Author = {Bryant, R}, Title = {Surfaces of mean curvature one in hyperbolic space}, Volume = {154-155}, Series = {Astérisque}, Pages = {321-347}, Booktitle = {Théorie des variétés minimales et applications (Palaiseau, 1983–1984)}, Publisher = {Société Mathématique de France}, Year = {1988}, MRNUMBER = {955072}, Key = {fds318278} } @article{fds243397, Author = {Bryant, RL}, Title = {Metrics with exceptional holonomy}, Journal = {Ann. of Math. (2)}, Volume = {126}, Number = {3}, Pages = {525-576}, Year = {1987}, MRNUMBER = {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 = {65-76}, Booktitle = {Differential geometry: the interface between pure and applied mathematics (San Antonio, Tex., 1986)}, Publisher = {American Mathematical Society}, Editor = {Luksic, M and Martin, C and Shadwick, W}, Year = {1987}, ISBN = {0-8218-5075-X}, MRNUMBER = {89f:58037}, Key = {fds318280} } @article{fds318281, Author = {Bryant, R}, Title = {A survey of Riemannian metrics with special holonomy groups}, Pages = {505-514}, Booktitle = {Proceedings of the International Congress of Mathematicians. Vol. 1, 2. (Berkeley, Calif., 1986)}, Publisher = {American Mathematical Society}, Editor = {Gleason, A}, Year = {1987}, ISBN = {0-8218-0110-4}, MRNUMBER = {89f:53068}, Key = {fds318281} } @article{fds318282, Author = {Bryant, R}, Title = {Minimal Lagrangian submanifolds of Kähler-Einstein manifolds}, Volume = {1255}, Series = {Lecture Notes in Math.}, Pages = {1-12}, Booktitle = {Differential geometry and differential equations (Shanghai, 1985)}, Publisher = {Springer Verlag}, Editor = {Gu, C and Berger, M and Bryant, RL}, Year = {1987}, ISBN = {3-540-17849-X}, MRNUMBER = {88j:53061}, 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 = {525-570}, Year = {1986}, MRNUMBER = {88a:58044}, Key = {fds243396} } @article{fds243395, Author = {Bryant, RL}, Title = {Lie groups and twistor spaces}, Journal = {Duke Mathematical Journal}, Volume = {52}, Number = {1}, Pages = {223-261}, Year = {1985}, Month = {March}, MRNUMBER = {87d:58047}, Doi = {10.1215/S0012-7094-85-05213-5}, Key = {fds243395} } @article{fds243394, Author = {Bryant, RL}, Title = {Minimal surfaces of constant curvature in S^n}, Journal = {Trans. Amer. Math. Soc.}, Volume = {290}, Number = {1}, Pages = {259-271}, Year = {1985}, MRNUMBER = {87c:53110}, Key = {fds243394} } @article{fds318283, Author = {Bryant, R}, Title = {Metrics with holonomy G2 or Spin(7)}, Volume = {1111}, Series = {Lecture Notes in Math.}, Pages = {269-277}, Booktitle = {Workshop Bonn 1984 (Bonn, 1984)}, Publisher = {SPRINGER}, Editor = {Hirzebruch, F and Schwermer, J and Suter, S}, Year = {1985}, MRNUMBER = {87a:53082}, Key = {fds318283} } @article{fds243393, Author = {Bryant, RL}, Title = {A duality theorem for Willmore surfaces}, Journal = {J. Differential Geom.}, Volume = {20}, Number = {1}, Pages = {23-53}, Year = {1984}, MRNUMBER = {86j:58029}, Key = {fds243393} } @article{fds243391, Author = {R. Bryant and Bryant, RL and Griffiths, PA and Yang, D}, Title = {Characteristics and existence of isometric embeddings}, Journal = {Duke Mathematical Journal}, Volume = {50}, Number = {4}, Pages = {893-994}, Year = {1983}, Month = {December}, MRNUMBER = {85d:53027}, Doi = {10.1215/S0012-7094-83-05040-8}, 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 = {803-892}, Year = {1983}, MRNUMBER = {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 = {77-102}, Booktitle = {Arithmetic and geometry, Vol. II}, Publisher = {Birkhäuser Boston}, Editor = {Artin, M and Tate, J}, Year = {1983}, ISBN = {3-7643-3133-X}, MRNUMBER = {86a:32044}, Key = {fds318284} } @article{fds243389, Author = {Bryant, RL}, 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. 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 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 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. The technique used is exterior differential systems together with the Chern-Moser 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 = {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 O or octonian space. The fact that the stabilizer of an oriented 2-plane in Spin(7) is U(3) implies that any oriented 6-manifold in O inherits a U(3)-structure. The first part of the paper studies the generality of the 6-manifolds whose inherited U(3)-structure is symplectic, complex, or Kähler, etc. by applying the theory of exterior differential systems. I then turn to the study of the standard 6-sphere in O as an almost complex manifold and study the space of what are now called pseudo-holomorphic curves in the 6-sphere. I prove that every compact Riemann surface occurs as a (possibly ramified) pseudo-holomorphic curve in the 6-sphere. I also show that all of the genus zero pseudo-holomorphic curves in the 6-sphere are algebraic as surfaces. Reprints are available.}, Key = {fds243390} } @article{fds243410, Author = {Bryant, RL}, Title = {Conformal and minimal immersions of compact surfaces into the 4-sphere}, Journal = {Journal of Differential Geometry}, Volume = {17}, Number = {3}, Pages = {455-473}, Year = {1982}, MRNUMBER = {84a:53062}, Doi = {10.4310/jdg/1214437137}, Key = {fds243410} } @article{fds318285, Author = {R. Bryant and Bryant, R and Chern, SS and Griffiths, PA}, Title = {Exterior Differential Systems}, Volume = {1}, Pages = {219-338}, Booktitle = {Proceedings of the 1980 Beijing Symposium on Differential Geometry and Differential Equations (Beijing, 1980)}, Publisher = {Science Press; Gordon & Breach Science Publishers}, Editor = {Chern, SS and Wu, WT}, Year = {1982}, ISBN = {0-677-16420-3}, MRNUMBER = {85k:58005}, 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 = {4657-4660}, Year = {1981}, MRNUMBER = {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}, Abstract = {For smooth manifolds equipped with various geometric structures, we construct complexes that replace the de Rham complex in providing an alternative fine resolution of the sheaf of locally constant functions. In case that the geometric structure is that of a parabolic geometry, our complexes coincide with the Bernstein- Gelfand-Gelfand complex associated with the trivial representation. However, at least in the cases we discuss, our constructions are relatively simple and avoid most of the machinery of parabolic geometry. Moreover, our method extends to certain geometries beyond the parabolic realm.}, Key = {fds216495} } %% Preprints @article{fds320294, Author = {Bryant, RL}, Title = {On the convex Pfaff-Darboux Theorem of Ekeland and Nirenberg}, Year = {2015}, Month = {December}, Abstract = {The classical Pfaff-Darboux Theorem, which provides local `normal forms' for 1-forms on manifolds, has applications in the theory of certain economic models. 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 Pfaff-Darboux Theorem. (The appropriate notion of `convexity' is a feature of the economic model. In the simplest case, when the economic model is formulated in a domain in n-space, convexity has its usual meaning. In 2002, Ekeland and Nirenberg were able to characterize necessary and sufficient conditions for a given 1-form 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 Pfaff-Darboux Theorem, one that covers the case of a Legendrian foliation in which the notion of convexity is defined in terms of a torsion-free affine connection on the underlying manifold. (The main result 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 2-tori}, Year = {2015}, Month = {July}, 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 almost-complex structures on the six-sphere}, Year = {2014}, Month = {May}, Keywords = {6-sphere • complex structure • exceptional geometry}, Abstract = {In 2003, S.-s. Chern began a study of almost-complex structures on the 6-sphere, with the idea of exploiting the special properties of its well-known almost-complex 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 almost-complex structure on the 6-sphere, he did prove a significant identity that resolves the question for an interesting class of almost-complex structures on the 6-sphere.}, Key = {fds320296} } @article{fds320297, Author = {Bryant, RL}, Title = {Notes on exterior differential systems}, Year = {2014}, Month = {May}, Keywords = {exterior differential systems • Lie theory • differential geometry}, Abstract = {These are notes for a very rapid introduction to the basics of exterior differential systems and their connection with what is now known as Lie theory, together with some typical and not-so-typical applications to illustrate their use.}, Key = {fds320297} } @article{fds225242, Author = {R.L. Bryant and Feng Xu}, 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 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}, Abstract = {A unimodular complex surface is a complex 2-manifold X endowed with a holomorphic volume form. A strictly pseudoconvex real hypersurface M in X inherits not only a CR-structure but a canonical coframing as well. In this article, this canonical coframing on M is defined, its invariants are discussed and interpreted geometrically, and its basic properties are studied. A natural evolution equation for strictly pseudoconvex real hypersurfaces in unimodular complex surfaces is defined, some of its properties are discussed, and several examples are computed. The locally homogeneous examples are determined and used to illustrate various features of the geometry of the induced structure on the hypersurface.}, Key = {fds320300} }