We obtain divergence theorems on the solution space of an elliptic stochastic differential equation defined on a smooth compact finite-dimensional manifold M . The resulting divergences are expressed in terms of the Ricci curvature of M with respect to a natural metric on M induced by the stochastic differential equation. The proofs of the main theorems are based on the lifting method of Mallia...

Given a Riemannian manifold (M, g), we study the solutions of heat equations associated with second order differential operators in divergence form that are uniformly elliptic with respect to g . Typical examples of such operators are the Laplace operators of Riemannian structures which are quasi-isometric to g . We first prove some Poincare and Sobolev inequalities on geodesic balls. Then we u...

This note studies Arveson’s curvature invariant for d-contractions T = (T1, T2, . . . , Td) for the special case d = 1, referring to a single contraction operator T on a Hilbert space. It establishes a formula which gives an easy-to-understand meaning for the curvature of a single contraction. The formula is applied to give an example of an operator with nonintegral curvature. Under the additio...

Abstract. We give sufficient conditions for a measured length space (X, d, ν) to admit local and global Poincaré inequalities. We first introduce a condition DM on (X, d, ν), defined in terms of transport of measures. We show that DM , together with a doubling condition on ν, implies a scale-invariant local Poincaré inequality. We show that if (X, d, ν) has nonnegative N -Ricci curvature and ha...

This paper develops a method to analyze and compute the lines of curvature and their differential geometry defined on implicit surfaces. With our technique, we can explicitly obtain the analytic formulae of the associated geometric attributes of an implicit surface, e.g. torsion of a line of curvature and Gaussian curvature. Additionally, it can be used to directly derive the closed formulae of...

Systolic complexes were introduced in Januszkiewicz–Świa̧tkowski [12] and, independently, in Haglund [10]. They are simply connected simplicial complexes satisfying a certain condition that we call simplicial nonpositive curvature (abbreviated SNPC). The condition is local and purely combinatorial. It neither implies nor is implied by nonpositive curvature for geodesic metrics on complexes, but ...

Our goal in this paper is to obtain further information about the curvature of gradient shrinking Ricci solitons. This is important for a better understanding and ultimately for the classification of these manifolds. The classification of gradient shrinkers is known in dimensions 2 and 3, and assuming locally conformally flatness, in all dimensions n ≥ 4 (see [14, 13, 6, 15, 20, 12, 2]). Many o...

The optimal transport problem has received the attention of many researchers in the last two decades, and its popularity is still increasing. This is mainly motivated by the discovery of unexpected connections between optimal transport and problems in physics, geometry, partial differential equations, etc. To give an example, consider the following geometric statement: Let (Mk, gk, volk) be a s...

The optimal transport problem has received the attention of many researchers in the last two decades, and its popularity is still increasing. This is mainly motivated by the discovery of unexpected connections between optimal transport and problems in physics, geometry, partial differential equations, etc. To give an example, consider the following geometric statement: Let (Mk, gk, volk) be a s...

We discuss the relation between ‘bare’ cosmological parameters as the true spatial average characteristics that determine the cosmological model, and the parameters interpreted by observers with a “Friedmannian bias”, i.e., within a homogeneous space geometry. We may say that the latter are ‘dressed’ by the smoothed–out geometrical inhomogeneities of the surveyed spatial region. We identify two...