# Robert Bryant

- Phillip Griffiths Professor of Mathematics
- Professor in the Department of Mathematics

**External address:**103 Physics Bldg, West Campus, Durham, NC 27708

**Internal office address:**Box 90320, Durham, NC 27708-0320

**Phone:**(919) 660-2817

**Office Hours:**

Tuesdays and Thursdays, 10:30-12:00PM, and by appointment

### Research Areas and Keywords

##### Algebra & Combinatorics

integrability, symplectic geometry

##### Analysis

differential geometry, exterior differential systems, complex geometry

##### Computational Mathematics

integrability

##### Geometry: Differential & Algebraic

differential geometry, holonomy, exterior differential systems, integrability, curvature, Lie groups, symplectic geometry, complex geometry, homology

##### Mathematical Physics

holonomy, exterior differential systems, symplectic geometry

##### PDE & Dynamical Systems

differential geometry, holonomy, exterior differential systems, integrability, symplectic geometry

##### Topology

curvature, Lie groups, homology

My research concerns problems in the geometric theory of partial differential equations. More specifically, I work on conservation laws for PDE, Finsler geometry, projective geometry, and Riemannian geometry, including calibrations and the theory of holonomy.

Much of my work involves or develops techniques for studying systems of partial differential equations that arise in geometric problems. Because of their built-in invariance properties, these systems often have special features that make them difficult to treat by the standard tools of analysis, and so my approach uses ideas and techniques from the theory of *exterior differential systems*, a collection of tools for analyzing such PDE systems that treats them in a coordinate-free way, focusing instead on their properties that are invariant under diffeomorphism or other transformations.

I’m particularly interested in geometric structures constrained by natural conditions, such as Riemannian manifolds whose curvature tensor satisfies some identity or that supports some additional geometric structure, such as a parallel differential form or other geometric structures that satisfy some partial integrability conditions and in constructing examples of such geometric structures, such as Finsler metrics with constant flag curvature.

I am also the Director of the Simons Collaboration Special Holonomy in Geometry, Analysis, and Physics, and a considerable focus of my research and that of my students is directed towards problems in this area.

Bryant, R. L. “Lie groups and twlstor spaces.” *Duke Mathematical Journal*, vol. 52, no. 1, Jan. 1985, pp. 223–61. *Scopus*, doi:10.1215/S0012-7094-85-05213-5.
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Bryant, R. L. “Minimal surfaces of constant curvature in sn.” *Transactions of the American Mathematical Society*, vol. 290, no. 1, Jan. 1985, pp. 259–71. *Scopus*, doi:10.1090/S0002-9947-1985-0787964-8.
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Bryant, R. L. “A duality theorem for willmore surfaces.” *Journal of Differential Geometry*, vol. 20, no. 1, Jan. 1984, pp. 23–53. *Scopus*, doi:10.4310/jdg/1214438991.
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Berger, E., et al. “The Gauss equations and rigidity of isometric embeddings.” *Duke Mathematical Journal*, vol. 50, no. 3, Jan. 1983, pp. 803–92. *Scopus*, doi:10.1215/S0012-7094-83-05039-1.
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Bryant, R. L., et al. “Characteristics and existence of isometric embeddings.” *Duke Mathematical Journal*, vol. 50, no. 4, Jan. 1983, pp. 893–994. *Scopus*, doi:10.1215/S0012-7094-83-05040-8.
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Bryant, R. L. “Conformal and minimal immersions of compact surfaces into the 4-sphere.” *Journal of Differential Geometry*, vol. 17, no. 3, Jan. 1982, pp. 455–73. *Scopus*, doi:10.4310/jdg/1214437137.
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Bryant, R. L. “Submanifolds and special structures on the octonians.” *Journal of Differential Geometry*, vol. 17, no. 2, Jan. 1982, pp. 185–232. *Scopus*, doi:10.4310/jdg/1214436919.
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Bryant, R. L. “Holomorphic curves in lorentzian cr-manifolds.” *Transactions of the American Mathematical Society*, vol. 272, no. 1, Jan. 1982, pp. 203–21. *Scopus*, doi:10.1090/S0002-9947-1982-0656486-4.
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Berger, E., and P. Griffiths. “Some isometric embedding and rigidity results for Riemannian manifolds.” *Proc. Nat. Acad. Sci. U.S.A.*, vol. 78, no. 8, 1981, pp. 4657–60.

Bryant, Robert L. “Recent Advances in the Theory of Holonomy.” *Seminaire Bourbaki*, vol. 1998, no. 99, pp. 351–74.
Open Access Copy