# Browsing by Author "Bryant, Robert"

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Item Open Access A circle quotient of a G2 cone(Differential Geometry and its Application, 2020-12-01) Bryant, Robert; Acharya, Bobby; Salamon, Simon© 2020 A study is made of R6 as a singular quotient of the conical space R+×CP3 with holonomy G2, with respect to an obvious action by U(1) on CP3 with fixed points. Closed expressions are found for the induced metric, and for both the curvature and symplectic 2-forms characterizing the reduction. All these tensors are invariant by a diagonal action of SO(3) on R6, which can be used effectively to describe the resulting geometrical features.Item Open Access Curvature homogeneous hypersurfaces in space forms(2024-04-02) Bryant, Robert; Florit, Luis; Ziller, WolfgangItem Open Access Exterior Differential Systems and Euler-Lagrange Partial Differential EquationsBryant, Robert; Griffiths, Phillip; Grossman, DanielWe use methods from exterior differential systems (EDS) to develop a geometric theory of scalar, first-order Lagrangian functionals and their associated Euler-Lagrange PDEs, subject to contact transformations. The first chapter contains an introduction of the classical Poincare-Cartan form in the context of EDS, followed by proofs of classical results, including a solution to the relevant inverse problem, Noether's theorem on symmetries and conservation laws, and several aspects of minimal hypersurfaces. In the second chapter, the equivalence problem for Poincare-Cartan forms is solved, giving the differential invariants of such a form, identifying associated geometric structures (including a family of affine hypersurfaces), and exhibiting certain "special" Euler-Lagrange equations characterized by their invariants. In the third chapter, we discuss a collection of Poincare-Cartan forms having a naturally associated conformal geometry, and exhibit the conservation laws for non-linear Poisson and wave equations that result from this. The fourth and final chapter briefly discusses additional PDE topics from this viewpoint--Euler-Lagrange PDE systems, higher order Lagrangians and conservation laws, identification of local minima for Lagrangian functionals, and Backlund transformations. No previous knowledge of exterior differential systems or of the calculus of variations is assumed.Item Open Access Flat metrics with a prescribed derived coframingBryant, Robert; Clelland, Jeanne NielsenThe following problem is addressed: A $3$-manifold $M$ is endowed with a triple $\Omega = (\Omega^1,\Omega^2,\Omega^3)$ of closed $2$-forms. One wants to construct a coframing $\omega= (\omega^1,\omega^2,\omega^3)$ of $M$ such that, first, $\mathrm{d}\omega^i = \Omega^i$ for $i=1,2,3$, and, second, the Riemannian metric $g=(\omega^1)^2+(\omega^2)^2+(\omega^3)^2$ be flat. We show that, in the `nonsingular case', i.e., when the three $2$-forms $\Omega^i_p$ span at least a $2$-dimensional subspace of $\Lambda^2(T^*_pM)$ and are real-analytic in some $p$-centered coordinates, this problem is always solvable on a neighborhood of $p\in M$, with the general solution $\omega$ depending on three arbitrary functions of two variables. Moreover, the characteristic variety of the generic solution $\omega$ can be taken to be a nonsingular cubic. Some singular situations are considered as well. In particular, we show that the problem is solvable locally when $\Omega^1,\Omega^2,\Omega^3$ are scalar multiples of a single 2-form that do not vanish simultaneously and satisfy a nondegeneracy condition. We also show by example that solutions may fail to exist when these conditions are not satisfied.Item Open Access Isadore M. Singer (1924–2021) In Memoriam Part 1: Scientific Works(Notices of the American Mathematical Society, 2022-10-01) Bryant, Robert; Bismut, Jean-Michel; Cheeger, Jeff; Griffiths, Phillip; Donaldson, Simon; Hitchin, Nigel; Lawson, H Blaine; Gromov, Michail; Marcus, Adam; Spielman, Daniel; Srivastava, Nikhil; Witten, EdwardItem Open Access Isadore M. Singer (1924–2021) In Memoriam Part 2: Personal Recollections(Notices of the American Mathematical Society, 2022-11-01) Bryant, Robert; Cheeger, Jeff; Griffiths, Phillip; Blum, Lenore; Burns, Dan; Connes, Alain; Donnelly, Harold; Ebin, David; Guillemin, Victor; Palais, Richard; Rossi, Hugo; Simons, James; Singer, Elliot; Singer, Natasha; Stanton, Nancy; Sternberg, ShlomoItem Open Access Masatake Kuranishi (1924–2021)(Notices of the American Mathematical Society, 2022-05-01) Phong, Duong H; Siu, Yum-Tong; Bryant, Robert; Chau, Albert; Falbel, Elisha; Fefferman, Charles; Friedman, Robert; Morgan, John; Futaki, Akito; Griffiths, Phillip; Kohn, Joseph J; Mok, Ngaiming; Mori, Shigefumi; Namba, Makoto; Noguchi, Junjiro; Ohsawa, Takeo; Sato, Mikio; Yau, Shing-TungItem Open Access Notes on Projective, Contact, and Null CurvesBryant, RobertThese are notes on some algebraic geometry of complex projective curves, together with an application to studying the contact curves in CP^3 and the null curves in the complex quadric Q^3 in CP^4, related by the well-known Klein correspondence. Most of this note consists of recounting the classical background. The main application is the explicit classification of rational null curves of low degree in Q^3. I have recently received a number of requests for these notes, so I am posting them to make them generally available.Item Open Access On the Convex Pfaff–Darboux Theorem of Ekeland and Nirenberg(Symmetry, Integrability and Geometry: Methods and Applications, 2023-08-23) Bryant, RobertItem Open Access The generality of closed G_2 solitons(Pure and Applied Mathematics Quarterly) Bryant, RobertItem Open Access The mathematical work of H. Blaine Lawson, Jr.(Pure and Applied Mathematics Quarterly, 2024-01-31) Bryant, Robert; Cheeger, Jeff; Lima-Filho, Paulo; Rosenberg, Jonathan; White, Brian