Locally symmetric immersions

Tight Equivariant Immersions of Symmetric Spaces

Introduction. Let G/K be a compact, irreducible symmetric space and Q = f+p the Lie algebra of G. If ir is a non trivial real class-one representation of G on E with O^e, infixed, then the map TlG/K —*E given by gK—*ir{g)e gives an immersion of G/K into E. The purpose of this note is to announce the classification of such immersions with minimal absolute curvature (i.e., are tight) [ l ] , [4]....

Minimal Immersions of Symmetric Spaces into Spheres

Isospectral locally symmetric manifolds

In this article we construct closed, isospectral, non-isometric locally symmetric manifolds. We have three main results. First, we construct arbitrarily large sets of closed, isospectral, non-isometric manifolds. Second, we show the growth of size these sets of isospectral manifolds as a function of volume is super-polynomial. Finally, we construct pairs of infinite towers of finite covers of a...

Counting Locally Symmetric Manifolds

We give quantitive estimates for the number of locally symmetric spaces of a given type with bounded volume. Explicitly, let S be a symmetric space of non-compact type without Euclidean de Rham factors. Then, after rescaling appropriately the Riemannian metric, the following hold. Theorem A If rank(S) = 1 and S ≇ H2,H3, then there are at most V V Riemannian manifolds, locally isometric to S, wi...

k-SYMMETRIC AKS SYSTEMS AND FLAT IMMERSIONS INTO SPHERES

We define a large class of integrable nonlinear PDE’s, k-symmetric AKS systems, whose solutions evolve on finite dimensional subalgebras of loop algebras, and linearize on an associated algebraic curve. We prove that periodicity of the associated algebraic data implies a type of quasiperiodicity for the solution, and show that the problem of isometrically immersing n-dimensional Euclidean space...