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.
Highlights
IntroductionAn identity map idM : M → M from a differentiable manifold M into itself, known as an identity transformation, is a map that always returns to the same point that was used in as its argument
The main purpose of this paper is to provide a survey on known results about identity maps from various Riemannian geometric points of view
We present results on the identity maps of tangent bundles of manifolds equipped with Sasakian, lift-complete, or g-natural metrics
Summary
An identity map idM : M → M from a differentiable manifold M into itself, known as an identity transformation, is a map that always returns to the same point that was used in as its argument. The author and Nagano proved in [2] that a Riemannian manifold (M, g) admits a geodesic vector field v if and only if the identity map idM : (M, g) → (M, Lvg) is a harmonic map, where Lv denotes the Lie derivative with respect to v. To study the conformal geometry of the identity map on a manifold M, we assume that domain M and range M equipped with metrics g and g , respectively. We present results on the identity maps of tangent bundles of manifolds equipped with Sasakian, lift-complete, or g-natural metrics. We discuss the identity maps on the tangent bundles of Walker 4-manifolds and of Gödel-type spacetimes
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