Affine images of Riemannian manifolds

Abstract

We describe all affine maps from a Riemannian manifold to a metric space and all possible image spaces.

Rigidity of Busemann convex Finsler metrics

“…Metric spaces affinely equivalent to Riemannian manifolds are characterized in [Lyt12] in a way similar to Szabó's metrization theorem. In fact, every such space is a limit of smooth Finsler metrics affinely equivalent to the same Riemannian one, see Theorem 1.4 and Lemma 4.1 in [Lyt12]. This and Corollary 1.3 imply an affirmative answer to the second question above if (X, d 0 ) is a smooth Finsler manifold.…”

Rigidity for the spectral gap on $RCD(K,\infty)$-spaces

“…The existence of a nonconstant affine function is a strong constraint and forces the space to possess some splitting phenomenon. See [In, Ma, AB, HL] for related results concerning affine functions on Riemannian manifolds or metric spaces, and [Oh1,Ly,BMS] for further studies on affine maps between (or into) metric spaces.…”

Affine maps between CAT(0) spaces

“…Similar examples can be obtained by replacing l 1 with a norm determined by a suitable centrally symmetric convex body. Indeed, this idea gives rise to the Finsler norms explored in [14].…”

“…If X and Y are Riemannian, the answer to our problem is classical [11]: any self-affine map of an irreducible, non-Euclidean Riemannian manifold is a dilation. Remarkably, Lytchak [14], following work by Ohta [18], classified affine maps from a Riemannian manifold X to any metric space Y as dilations, as long as X is not a product or a higher rank symmetric space. In the latter cases, they produce counterexamples by endowing these spaces with suitable Finsler metrics.…”

“…Matveev has obtained strong positive results, especially for closed Riemannian manifolds with negative curvature [16].If X and Y are Riemannian, the answer to our problem is classical [11]: any self-affine map of an irreducible, non-Euclidean Riemannian manifold is a dilation. Remarkably, Lytchak [14], following work by Ohta [18], classified affine maps from a Riemannian manifold X to any metric space Y as dilations, as long as X is not a product or a higher rank symmetric space. In the latter cases, they produce counterexamples by endowing these spaces with suitable Finsler metrics.Lytchak and Schröder [15], and later Hitzelberger and Lytchak [9] further investigated the case of realvalued functions on a CAT(κ) metric space, and obtained severe restrictions.…”