Robert L. Bryant

Lie groups and twistor spaces

On considere un espace symetrique de Riemann simplement connexe M, la connexion de Levi-Civita ⊇ et le fibre torseur metrique #7B-F(M)⊆J~(M) dont les fibres #7B-F x (M) pour x∈M est constitue de structures complexes orthogonales

A solution of a problem of Sophus Lie: normal forms of two-dimensional metrics admitting two projective vector fields

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.

Some Observations on the Infinitesimal Period Relations for Regular Threefolds with Trivial Canonical Bundle

It is well-known that, aside from algebraic curves, abelian varieties, and a few other isolated cases such as K3 surfaces, the period matrices of a family of algebraic varieties satisfy non-trivial universal infinitesimal period relations. In this note we shall discuss some remarkable properties of any local solution to the differential system given by the infinitesimal period relation associated to polarized Ilodge structures of weight three with Ilodge number h 3,0 = 1.

Some remarks on the geometry of austere manifolds

We prove several structure theorems about the special class of minimal submanifolds which Harvey and Lawson have called “austere” and which arose in connection with their foundational work on calibrations. The condition of austerity is a pontwise condition on the second fundamental form and essentially requires that the non-zero eigenvalues of the second fundamental form in any normal direction at any point occur in oppositely signed pairs. We solve the pointwise problem of describing the set of austere second fundamental forms in dimension at most four and the local problem of describing the austere three-folds in Euclidean space in all dimensions.

Calibrated embeddings in the special Lagrangian and coassociative cases

Every closed, oriented, real analytic Riemannian3–manifold can be isometrically embedded as a specialLagrangian submanifold of a Calabi–Yau 3–fold, even as thereal locus of an antiholomorphic, isometric involution. Every closed,oriented, real analytic Riemannian 4–manifold whose bundle of self-dual2–forms is trivial can be isometrically embedded as a coassociativesubmanifold in a G2-manifold, even as the fixed locus of ananti-G2 involution.These results, when coupledwith McLeans analysis of the moduli spaces of such calibratedsubmanifolds, yield a plentiful supply of examples of compact calibratedsubmanifolds with nontrivial deformation spaces.

The Area Derivative of a Space-Filling Diagram

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.

Hyperbolic exterior differential systems and their conservation laws, part II

This research was supported in part by NSF Grant DMS 9205222 (Bryant), an NSERC Postdoctoral Fellowship (Hsu), and the Institute for Advanced Study (Griffiths and Hsu).IAS Preprint: 1/25/94. Part II will appear in Vol. 1, No. 2 of this journal.

D-branes and Spinc structures

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 supersymmetric cycle is automatically Spin c .

Submanifolds in hyper-Kähler geometry

A calibration q is a differential form on a Riemannian manifold with two additional properties. First, the form should be closed under exterior differentiation. Second, it should be less than or equal to the volume form on each oriented submanifold (of the same dimension as the degree of the form sb). Each calibration 0 determines a geometry of submanifolds, namely those oriented submanifolds for which 0 restricts to be exactly the volume form. Such submanifolds are called 0-submanifolds. The Fundamental Lemma of the theory of calibrations says that each 0-submanifold is homologically area minimizing. A Kahler form provides the most important classical example of a calibration. In this case the 0-submanifolds are just the complex submanifolds of dimension one. One of the most interesting nonclassical examples of a calibration, introduced in Harvey and Lawson [HL], is a 4-form (I on euclidean R8 called the Cayley 4-form. This 4-form 1 has an elegant description in terms of the algebra of octonians 0, and is fixed by the subgroup Spin(7) of the group of all orthogonal transformations on 0. As such, it would appear unlikely that (I would have higher dimensional generalizations. The purpose of this paper is to provide the higher dimensional analogue. The Cayley form (I on R8 _ 0 can also be considered in a very natural way as a 4-form on H 2, the quaternionic plane. After choosing to distinguish the scalar quaternion K, the 4-form can be expressed as

Linear Differential Systems

The goal of this chapter is to develop the formalism of linear Pfaffian differential systems in a form that will facilitate the computation of examples.