Instability of the Liouville property for quasi-isometric Riemannian manifolds and reversible Markov chains

Isometric Embeddings of Riemannian Manifolds

The dot in (1) denotes the usual scalar product of R. The notion embedding means, that w is locally an immersion and globally a homeomorphism of M onto the subspace u(M) of R*. If an embedding w : M -• R satisfies (1) on the whole M, we speak of an isometric embedding. If w is an immersion and a solution of (1) in a (possibly small) neighbourhood of any point of M, we speak of a local isometric...

Isometric Embedding of Riemannian Manifolds

Relative Isometric Embeddings of Riemannian Manifolds

We prove the existence of C1 isometric embeddings, and C∞ approximate isometric embeddings, of Riemannian manifolds into Euclidean space with prescribed values in a neighborhood of a point.

Some isometric embedding and rigidity results for Riemannian manifolds.

In this note, we announce some new results concerning rigidity and isometric embeddings of Riemannian manifolds. A special case of the main result states that a general M(n) [unk] R(n+r), r</=(n - 1)(n - 2)/2, is uniquely determined up to finitely many constants by its induced ds(2).

weak-reversible Markov chains

The theory of L-spectral gaps for reversible Markov chains has been studied by many authors. In this paper we consider positive recurrent general state space Markov chains with stationary transition probabilities. Replacing the assumption of reversibility by a less strong one, we still obtain a simple necessary and sufficient condition for the spectral gap property of the associated Markov oper...