# Seminar talk, 24 February 2016

Speaker: Valery Yumaguzhin

Title: Differential invariants and exact solutions of the Einstein and the Einstein-Maxwell equations

Abstract:

The talk will discuss the results of [1] and [2].

It will be shown that on an arbitrary solution of the Einstein equation an eigenvalue of the Weyl operator gives rise to hyperbolic and elliptic distributions, and on a general position solution of Einstein-Maxwell equation electromagnetic tensor gives rise to hyperbolic and elliptic distributions.

A solution of the Einstein or the Einstein-Maxwell equation is called completely geodesic if these distributions on it are completely integrable and the maximal integral manifolds are completely geodesic.

In the talk for both equations we will compute an explicit family of completely geodesic solutions parametrized by constants, two functions in one variable, and one harmonic function.

References:

[1] V. Lychagin, V. Yumaguzhin, Differential invariants and exact solutions of the Einstein-Maxwell equation, Anal. Math. Phys., 2016, to appear.

[2] V. Lychagin, V. Yumaguzhin, Differential invariants and exact solutions of the Einstein equation, Anal. Math. Phys., 2016, to appear.