# Browsing by Author "Bryant, R"

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Item Open Access An introduction to Lie groups and symplectic geometry(Geometry and quantum field theory (Park City, UT, 1991), 1995) Bryant, RItem Open Access Complex analysis and a class of Weingarten surfaces(2011-05-27) Bryant, RAn idea of Hopf's for applying complex analysis to the study of constant mean curvature spheres is generalized to cover a wider class of spheres, namely, those satisfying a Weingarten relation of a certain type, namely H = f(H^2-K) for some smooth function f, where H and K are the mean and Gauss curvatures, respectively. The results are either not new or are minor extensions of known results, but the method, which involves introducing a different conformal structure on the surface than the one induced by the first fundamental form, is different from the one used by Hopf and requires less technical results from the theory of PDE than Hopf's methods. This is a TeXed version of a manuscript dating from early 1984. It was never submitted for publication, though it circulated to some people and has been referred to from time to time in published articles. It is being provided now for the convenience of those who have asked for a copy. Except for the correction of various grammatical or typographical mistakes and infelicities and the addition of some (clearly marked) comments at the end of the introduction, the text is that of the original.Item Open Access Élie Cartan and geometric duality(Journées Élie Cartan 1998 et 1999, 2000) Bryant, RItem Open Access Geodesic behavior for Finsler metrics of constant positive flag curvature on S^2(Journal of Differential Geometry) Bryant, R; Foulon, P; Ivanov, S; Matveev, VS; Ziller, WWe study non-reversible Finsler metrics with constant flag curvature 1 on S^2 and show that the geodesic flow of every such metric is conjugate to that of one of Katok's examples, which form a 1-parameter family. In particular, the length of the shortest closed geodesic is a complete invariant of the geodesic flow. We also show, in any dimension, that the geodesic flow of a Finsler metrics with constant positive flag curvature is completely integrable. Finally, we give an example of a Finsler metric on S^2 with positive flag curvature such that no two closed geodesics intersect and show that this is not possible when the metric is reversible or have constant flag curvatureItem Open Access Geodesically reversible Finsler 2-spheres of constant curvature(Inspired by S. S. Chern---A Memorial Volume in Honor of a Great Mathematician, 2006) Bryant, RA 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.Item Open Access Holonomy and Special Geometries(Dirac Operators: Yesterday and Today, 2005) Bryant, RItem Open Access Laplacian Flow for Closed $G_2$-Structures: Short Time Behavior(2011-01-11) Bryant, R; Xu, FWe 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.Item Open Access Metrisability of two-dimensional projective structures(Journal of Differential Geometry, 2009-12-01) Bryant, R; Dunajski, M; Eastwood, MWe carry out the programme of R. Liouville [19] to construct an explicit local obstruction to the existence of a Levi-Civita connection within a given projective structure [Γ] on a surface. The obstruction is of order 5 in the components of a connection in a projective class. It can be expressed as a point invariant for a second order ODE whose integral curves are the geodesics of [Γ] or as a weighted scalar projective invariant of the projective class. If the obstruction vanishes we find the sufficient conditions for the existence of a metric in the real analytic case. In the generic case they are expressed by the vanishing of two invariants of order 6 in the connection. In degenerate cases the sufficient obstruction is of order at most 8.Item Open Access On extremals with prescribed Lagrangian densities(Manifolds and geometry (Pisa, 1993), 1996) Bryant, RThis 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.Item Open Access Pseudo-Reimannian metrics with parallel spinor fields and vanishing Ricci tensor(Global analysis and harmonic analysis (Marseille-Luminy, 1999), 2000) Bryant, RI 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).Item Open Access Some remarks on G2-structures(Proceedings of Gökova Geometry-Topology Conference 2005, 2006) Bryant, RThis article consists of loosely related remarks about the geometry of G2-structures on 7-manifolds, some of which are based on unpublished joint work with two other people: F. Reese Harvey and Steven Altschuler. After some preliminary background information about the group G2 and its representation theory, a set of techniques is introduced for calculating the differential invariants of G2-structures and the rest of the article is applications of these results. Some of the results that may be of interest are as follows: First, a formula is derived for the scalar curvature and Ricci curvature of a G2-structure in terms of its torsion and covariant derivatives with respect to the ‘natural connection’ (as opposed to the Levi-Civita connection) associated to a G2-structure. When the fundamental 3-form of the G2-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. These formulae are also used to generalize a recent result of Cleyton and Ivanov [3] about the nonexistence of closed Einstein G2-structures (other than the Ricci-flat ones) on compact 7-manifolds to a nonexistence result for closed G2-structures whose Ricci tensor is too tightly pinched. Second, some discussion is given of the geometry of the first and second order invariants of G2-structures in terms of the representation theory of G2. Third, 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. Some of this work was subsumed in the work of Hitchin [12] and Joyce [14]. 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.Item Open Access The origins of spectra, an organization for LGBT mathematicians(Notices of the American Mathematical Society, 2019-06-01) Bryant, R; Buckmire, R; Khadjavi, L; Lind, D