# Invariant Ricci-flat K\"ahler metrics on tangent bundles of compact symmetric spaces

## TL;DR

This paper classifies all G-invariant Ricci-flat Kähler metrics on the tangent bundles of compact symmetric spaces, providing explicit descriptions and new examples, including a family of metrics on the 2-sphere.

## Contribution

It offers a comprehensive description of invariant Ricci-flat Kähler metrics on tangent bundles of symmetric spaces, introducing new metrics and explicit classifications.

## Key findings

- Explicit description of all G-invariant Ricci-flat Kähler metrics on tangent bundles.
- Identification of a new one-parameter family of metrics on T S^2.
- Includes known Eguchi-Hanson-Stenzel metrics as special cases.

## Abstract

We give a description of all $G$-invariant Ricci-flat K\"ahler metrics on the canonical complexification of any compact Riemannian symmetric space $G/K$ of arbitrary rank, by using some special local $(1,0)$ vector fields on $T(G/K)$. As the simplest application, we obtain the explicit description of the set of all complete $\mathrm{SO}(3)$-invariant Ricci-flat K\"ahler metrics on $T{\mathbb S}^2$, which includes the well-known Eguchi-Hanson-Stenzel metrics and a new one-parameter family of metrics.