Second order Einstein deformations

TL;DR

This paper investigates the second order integrability of Einstein deformations on compact Riemannian and Kähler manifolds, providing new conditions, simplifying cases, and explicit classifications, including rigidity results.

Contribution

It introduces a new expression for second order integrability conditions, simplifies the Kähler case, and fully characterizes second order integrable Einstein deformations on the complex 2-plane Grassmannian.

Findings

Second order integrability condition expressed in a new, compact form.

Complete classification of second order integrable Einstein deformations on the complex 2-plane Grassmannian.

Rigidity of the symmetric Einstein metric on Grassmannians for odd dimensions.

Abstract

We study the integrability to second order of infinitesimal Einstein deformations on compact Riemannian and in particular on K\"ahler manifolds. We find a new way of expressing the necessary and sufficient condition for integrability to second order, which also gives a very clear and compact way of writing the Koiso obstruction. As an application we consider the K\"ahler case, where the condition can be further simplified and in complex dimension $3$ turns out to be purely algebraic. One of our main results is the complete and explicit description of infinitesimal Einstein deformation integrable to second order on the complex $2$-plane Grassmannian, which also has a quaternion K\"ahler structure. As a striking consequence we find that the symmetric Einstein metric on the Grassmannian $ \mathrm{Gr}_2(\bbC^{n+2})$ for $n$ odd is rigid.