N-dimensional Compact Riemannian Manifold Research Articles

An Estimate Near the Boundary for the Spectral Function of the Laplace Operator

The main result of this paper is an estimate as in the title, near a sufficiently smooth part of the boundary of a compact n-dimensional Riemannian manifold Q, for either Dirichlet or Neumann boundary conditions. This yields a sharp estimate for the remainder in the Weyl asymptotic formula, for any boundary conditions. For manifolds whose boundaries have singularities such as edges or corners, the estimate remains valid with Dirichlet conditions.

On equivariant isometric embeddings

The representation p can be regarded as a Lie group homomorphism from G into the orthogonal group O(N) which acts on IE N by rotations and reflections; a smooth map X: M ~ I E N is equivariant with respect to p if and only if X(~p) =p(cr) X(p), for all ~r~G, pEM. The main theorem is true in both the C ~ and real analytic categories. We will work in the C ~ category for the time being, and return to the real analytic case in w 4. Moreover, the theorem holds for manifolds with boundary. The main analytic tool used by Nash to prove his isometric embedding theorem is an implicit function theorem based upon the Newton iteration method. The implicit function theorem applies to the equivariant case with virtually no change. In order to apply the implicit function theorem we need to approximate a given G-invariant Riemannian metric on M by a metric induced by an equivariant embedding; we will do this by using the theory of the Laplace operator on compact Riemannian manifolds. According to Gromov and Rokhlin [7], any n-dimensional compact Riemannian manifold can be isometrically embedded in IF, N, where N = (1/2) n(n + 1) + 3 n + 5. No such universal bound is possible in the equivariant case, and in fact, given any positive integer N, it is possible to construct a left invariant

Holonomy Groups Of Hypersurfaces

The restricted homogeneous holonomy group of an n–dimensional Riemannian manifold is a connected closed subgroup of the proper orthogonal group SO(n) [1]. In this note we shall prove that the restricted homogeneous holonomy group of an n-dimensional compact hypersurface in the Euclidean space is actually the proper orthogonal group SO(n) itself. This gives a necessary (of course, not sufficient) condition for the imbedding of an n-dimensional compact Riemannian manifold into the (n +1)–dimensional Euclidean space.