The notion of dually flat Finsler metrics arise from information geometry. In this paper, we will study a special class of Finsler metrics called Randers metrics to be dually flat. A simple characterization is provided and some non-trivial explicit examples are constructed. In particular, We will show that the dual flatness of a Randers metric always arises from that of some Riemannian metric b...

A method, due tó Elie Cartan, is used to give an algebraic classification of the non-reductive homogeneous pseudo-Riemannian manifolds of dimension four. Only one case with Lorentz signature can be Einstein without having constant curvature, and two cases with (2,2) signature are Einstein of which one is Ricci-flat. If a four-dimensional non-reductive homogeneous pseudo-Riemannian manifold is s...

In computational anatomy, one needs to perform statistics on shapes and transformations, and to transport these statistics from one geometry (e.g. a given subject) to another (e.g. the template). The geometric structure that appeared to be the best suited so far was the Riemannian setting. The statistical Riemannian framework was indeed pretty well developped for finite-dimensional manifolds an...

We use persistence modules and their corresponding barcodes to quantitatively distinguish between different fiberwise star-shaped domains in the cotangent bundle of a fixed manifold. The distance two is measured by nonlinear version classical Banach–Mazur distance, called symplectic denoted [Formula: see text]. relevant come from filtered homology are stable with respect Our main focus on space...

We classify all smooth flat Riemannian metrics on the two-dimensional plane. In complete case, it is well known that these are isometric to Euclidean metric. incomplete there an abundance of naturally occurring, non-isometric relevant and useful. Remarkably, study classification plane—as a subject—is new literature. Much our research focuses conformal form \(e^{2\varphi }g_0\), where \(\varphi ...

This paper constructs a class of complete Kähler metrics of positive holomorphic sectional curvature on C and finds that the constructed metrics satisfy the following properties: As the geodesic distance ρ → ∞, the volume of geodesic balls grows like O(ρ 2(β+1)n β+2 ) and the Riemannian scalar curvature decays like O(ρ − 2(β+1) β+2 ), where β ≥ 0.