Differential Geometry of Identity Maps: A Survey

Abstract

An identity map idM:M→M is a bijective map from a manifold M onto itself which carries each point of M return to the same point. To study the differential geometry of an identity map idM:M→M, we usually assume that the domain M and the range M admit metrics g and g′, respectively. The main purpose of this paper is to provide a comprehensive survey on the differential geometry of identity maps from various differential geometric points of view.

The Scalar Curvature of a Riemannian Almost Paracomplex Manifold and Its Conformal Transformations

“…An identity map id : M → M from a differentiable manifold M into itself, also known as an identity transformation, is defined as the map with domain and range M, which satisfies id(x) = x for any x ∈ M, and it is the simplest map, which is both continuous and bijective (see [15]). Here, we will consider the conformal geometry of the identity map on a manifold M, and we assume that the domain M and the range M of id are equipped with metrics g and ḡ, respectively.…”