Riemannian manifolds have been widely employed for video representations in visual classification tasks including videobased face recognition. The success mainly derives from learning a discriminant Riemannian metric which encodes the non-linear geometry of the underlying Riemannian manifolds. In this paper, we propose a novel metric learning framework to learn a distance metric across a Euclid...

We construct a natural L-metric on the perturbed Seiberg-Witten moduli spaces Mμ+ of a compact 4-manifold M , and we study the resulting Riemannian geometry of Mμ+ . We derive a formula which expresses the sectional curvature of Mμ+ in terms of the Green operators of the deformation complex of the Seiberg-Witten equations. In case M is simply connected, we construct a Riemannian metric on the S...

We give an account on Otto’s geometrical heuristics for realizing, on a compact Riemannian manifold M , the L Wasserstein distance restricted to smooth positive probability measures, as a Riemannian distance. The Hilbertian metric discovered by Otto is obtained as the base metric of a Riemannian submersion with total space, the group of diffeomorphisms of M equipped with the Arnol’d metric, and...

In the first part of this paper, some theorems are given for a Riemannian manifold with semi-symmetric metric connection. In the second part of it, some special vector fields, for example, torse-forming vector fields, recurrent vector fields and concurrent vector fields are examined in this manifold. We obtain some properties of this manifold having the vectors mentioned above.

The book [7] emphasizes three relevant aspects: first, the fact that the notion of convexity is strongly metric-dependent either through geodesics or through the Riemannian connection; second that Riemannian convexity of functions is a coordinate-free concept, and consequently it can be easily connected with symbolic computation; third, that the Riemannian structure is involved essentially in f...

We study how convergence of an observer whose state lives in a copy of the given system’s space can be established using a Riemannian metric. We show that the existence of an observer guaranteeing the property that a Riemannian distance between system and observer solutions is nonincreasing implies that the Lie derivative of the Riemannian metric along the system vector field is conditionally n...

The notion of Poisson manifold with compatible pseudo-metric was introduced by the author in [1]. In this paper, we introduce a new class of Lie algebras which we call a pseudo-Riemannian Lie algebras. The two notions are strongly related: we prove that a linear Poisson structure on the dual of a Lie algebra has a compatible pseudo-metric if and only if the Lie algebra is a pseudo-Riemannian Li...

A geometric conception is a method of a geometry construction. The Riemannian geometric conception and a new T-geometric one are considered. T-geometry is built only on the basis of information included in the metric (distance between two points). Such geometric concepts as dimension, manifold, metric tensor, curve are fundamental in the Riemannian conception of geometry, and they are derivativ...

Riemannian first-passage percolation (FPP) is a continuum analogue of standard FPP on the lattice, where the discrete passage times of standard FPP are replaced by a random Riemannian metric. We prove a shape theorem for this model— that balls in this metric grow linearly in time—and from this conclude that the metric is complete.