After recalling some background, we define Riemannian metrics and Riemannian manifolds. We analyze the basic tensorial operations that become available in the presence of a Riemannian metric. Then we construct the Levi-Civita connection, which is the basic " new " differential operator coming from such a metric. Background The purpose of this section is two–fold. On the one hand, we want to rel...

In this paper, we are concerned with the existence of local isometric embeddings into Euclidean space for analytic Riemannian metrics g , defined on a domain U ? R n which singular in sense that determinant metric tensor is allowed to vanish at an isolated point (say origin). Specifically, show that, under suitable technical assumptions, there exists embedding u from ( ? ? ? ) E 2 + 3 ? 4 / whe...

Let (N, J) be a simply connected 2n-dimensional nilpotent Lie group endowed with an invariant complex structure. We define a left invariant Riemannian metric on N compatible with J to be minimal, if it minimizes the norm of the invariant part of the Ricci tensor among all compatible metrics with the same scalar curvature. In [7], J. Lauret proved that minimal metrics (if any) are unique up to i...

Methods of estimating (Riemannian and Finsler) filling volumes by using nonexpanding maps to Banach spaces of L∞-type are developed and generalized. For every Finsler volume functional (such as the Busemann volume or the Holmes– Thompson volume), a natural extension is constructed from the class of Finsler metrics to all Lipschitz metrics, and the notion of area is defined for Lipschitz surface...

We prove the Riemannian Penrose conjecture, an important case of a conjecture made by Roger Penrose in 1973, by defining a new flow of metrics. This flow of metrics stays inside the class of asymptotically flat Riemannian 3-manifolds with nonnegative scalar curvature which contain minimal spheres. In particular, if we consider a Riemannian 3-manifold as a totally geodesic submanifold of a space...