The ciconia metric on the tangent bundle of an almost-Hermitian manifold

TL;DR

This paper introduces a new class of invariant metrics on the tangent bundle of almost-Hermitian manifolds, with specific applications to Riemannian surfaces that produce Ricci-flat Kähler manifolds in four dimensions.

Contribution

It defines the ciconia metric, a novel invariant metric on tangent bundles, and explores its properties, especially in the context of Riemannian surfaces.

Findings

New invariant metrics on tangent bundles of almost-Hermitian manifolds

Construction of Ricci-flat Kähler manifolds in four dimensions

Examples of Kählerian Ricci-flat manifolds from Riemannian surfaces

Abstract

We find a new class of invariant metrics existing on the tangent bundle of any given almost-Hermitian manifold. We focus here on the case of Riemannian surfaces, which yield new examples of K\"ahlerian Ricci-flat manifolds in four real dimensions.