Results 1 
9 of
9
GriffithsHarris Rigidity of Compact Hermitian Symmetric Spaces
"... I prove that any complex manifold that has a projective second fundamental form isomorphic to one of a rank two compact Hermitian symmetric space (other than a quadric hypersurface) at a general point must be an open subset of such a space. This contrasts the nonrigidity of all other compact Hermit ..."
Abstract

Cited by 8 (3 self)
 Add to MetaCart
I prove that any complex manifold that has a projective second fundamental form isomorphic to one of a rank two compact Hermitian symmetric space (other than a quadric hypersurface) at a general point must be an open subset of such a space. This contrasts the nonrigidity of all other compact Hermitian symmetric spaces observed in [12, 13]. A key step is the use of higher order Bertini type theorems that may be of interest in their own right.
Differential geometry of submanifolds of projective space
 2006 IMA WORKSHOP “SYMMETRIES AND OVERDETERMINED SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS"
, 2006
"... These are lecture notes on the rigidity of submanifolds of projective space “resembling” compact Hermitian symmetric spaces in their homogeneous embeddings. The ..."
Abstract

Cited by 5 (2 self)
 Add to MetaCart
These are lecture notes on the rigidity of submanifolds of projective space “resembling” compact Hermitian symmetric spaces in their homogeneous embeddings. The
FUBINI’S THEOREM IN CODIMENSION TWO
"... Abstract. We classify codimension two analytic submanifolds of projective space X n ⊂ CP n+2 having the property that any line through a general point x ∈ X having contact to order two with X at x automatically has contact to order three. We give applications to the study of the Debarrede Jong conj ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
Abstract. We classify codimension two analytic submanifolds of projective space X n ⊂ CP n+2 having the property that any line through a general point x ∈ X having contact to order two with X at x automatically has contact to order three. We give applications to the study of the Debarrede Jong conjecture and of varieties whose Fano variety of lines has dimension 2n − 4. 1.
THE ADJOINT VARIETY OF SLm+1C IS RIGID TO ORDER THREE
, 2006
"... Abstract. I prove that the adjoint variety of SLm+1C in P(slm+1C) is rigid to order three. The principle result of this paper is Theorem 4.6 (page 8) which asserts that the adjoint variety of the simple Lie group SLm+1C is rigid to order three. The result is extrinsic; roughly speaking, if a variety ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
Abstract. I prove that the adjoint variety of SLm+1C in P(slm+1C) is rigid to order three. The principle result of this paper is Theorem 4.6 (page 8) which asserts that the adjoint variety of the simple Lie group SLm+1C is rigid to order three. The result is extrinsic; roughly speaking, if a variety Y ⊂ P(slm+1C) = P m2 +2m−1, of dimension n = 2m − 1, resembles the adjoint variety to third order at a 3general point y ∈ Y, then there is a transformation in GL m 2 +2mC mapping Y onto the adjoint variety. The conclusion is significant because it is the first rigidity result for a variety with nonvanishing Fubini cubic F3 (a third order invariant). And it is striking that this is the first example of kth order rigidity for which the (k + 1)th order Fubini invariant is nonzero: F4 can not be normalized to zero. The proof is based on the E. Cartan’s method of moving frames. The reader may find similar applications of the technique to the study of submanifolds of CP N in [4, 9, 10, 11, 12], and their references. The paper is organized as follows §1 Notation is set. The firstorder adapted frame bundle associated to a variety is introduced, and the relative differential invariants Fk, or Fubini forms, are discussed. (These invariants
EXTERIOR DIFFERENTIAL SYSTEMS, LIE ALGEBRA COHOMOLOGY, AND THE RIGIDITY OF HOMOGENOUS VARIETIES
"... Abstract. These are expository notes from the 2008 Srni Winter School. They have two purposes: (1) to give a quick introduction to exterior differential systems (EDS), which is a collection of techniques for determining local existence to systems of partial differential equations, and (2) to give an ..."
Abstract
 Add to MetaCart
Abstract. These are expository notes from the 2008 Srni Winter School. They have two purposes: (1) to give a quick introduction to exterior differential systems (EDS), which is a collection of techniques for determining local existence to systems of partial differential equations, and (2) to give an exposition of recent work (joint with C. Robles) on the study of the FubiniGriffithsHarris rigidity of rational homogeneous varieties, which also involves an advance in the EDS technology.
DIFFERENTIAL GEOMETRY OF SUBMANIFOLDS OF PROJECTIVE SPACE
, 2006
"... Abstract. These are lecture notes on the rigidity of submanifolds of projective space “resembling” compact Hermitian symmetric spaces in their homogeneous embeddings. The results of ..."
Abstract
 Add to MetaCart
Abstract. These are lecture notes on the rigidity of submanifolds of projective space “resembling” compact Hermitian symmetric spaces in their homogeneous embeddings. The results of
1rigidity of CR submanifolds in spheres
, 2005
"... Abstract We propose a unified computational framework for the problem of deformation and rigidity of submanifolds in a homogeneous space under geometric constraint. A notion of 1rigidity of a submanifold under admissible deformations is introduced. It means every admissible deformation of the subma ..."
Abstract
 Add to MetaCart
Abstract We propose a unified computational framework for the problem of deformation and rigidity of submanifolds in a homogeneous space under geometric constraint. A notion of 1rigidity of a submanifold under admissible deformations is introduced. It means every admissible deformation of the submanifold osculates a one parameter family of motions up to 1st order. We implement this idea to the question of rigidity of CR submanifolds in spheres. A class of submanifolds called Bochner rigid submanifolds are shown to be 1rigid under type preserving CR deformations. 1rigidity is then extended to a rigid neighborhood theorem, which roughly states that if a CR submanifold M is Bochner rigid, then any pair of mutually CR equivalent CR submanifolds that are sufficiently close to M are congruent by an automorphism of the sphere. A local characterization of Whitney submanifold is obtained, which is an example of a CR submanifold that is not 1rigid. As a by product, we give a simple characterization of the proper holomorphic maps from the unit ball B n+1 to B 2n+1.
EXTERIOR DIFFERENTIAL SYSTEMS, LIE ALGEBRA COHOMOLOGY, AND THE RIGIDITY OF HOMOGENOUS VARIETIES
, 802
"... Abstract. These are expository notes from the 2008 Srni Winter School. They have two purposes: (1) to give a quick introduction to exterior differential systems (EDS), which is a collection of techniques for determining local existence to systems of partial differential equations, and (2) to give an ..."
Abstract
 Add to MetaCart
Abstract. These are expository notes from the 2008 Srni Winter School. They have two purposes: (1) to give a quick introduction to exterior differential systems (EDS), which is a collection of techniques for determining local existence to systems of partial differential equations, and (2) to give an exposition of recent work (joint with C. Robles) on the study of the FubiniGriffithsHarris rigidity of rational homogeneous varieties, which also involves an advance in the EDS technology.