A new geometric structure on tangent bundles

TL;DR

This paper introduces a new scalar flat metric on tangent bundles of Riemannian manifolds, explores its conformal flatness and symmetry properties, and studies submanifold conditions including geodesics and minimal Lagrangian graphs, with applications to the space of lines in 3D.

Contribution

It constructs a novel scalar flat metric on tangent bundles and characterizes its geometric properties, including conformal flatness and symmetry, along with submanifold and minimality conditions.

Findings

G constructs scalar flat metric G on TN.

G is locally conformally flat iff N is 2D or a space form.

G is locally symmetric iff g is locally symmetric.

Abstract

For a Riemannian manifold $(N,g)$, we construct a scalar flat metric $G$ in the tangent bundle $TN$. It is locally conformally flat if and only if either, $N$ is a 2-dimensional manifold or, $(N,g)$ is a real space form. It is also shown that $G$ is locally symmetric if and only if $g$ is locally symmetric. We then study submanifolds in $TN$ and, in particular, find the conditions for a curve to be geodesic. The conditions for a Lagrangian graph to be minimal or Hamiltonian minimal in the tangent bundle $T{\mathbb R}^n$ of the Euclidean real space ${\mathbb R}^n$ are studied. Finally, using the cross product in ${\mathbb R}^3$ we show that the space of oriented lines in ${\mathbb R}^3$ can be minimally isometrically embedded in $T{\mathbb R}^3$.