S. E. Stepanov, J. Mikeš, “Differential equations and their general solutions, as well as nonexistence theorems for six invariant classes of the vacuum constraint equations”, TMF, 2025, Volume 225, Number 1,Pages <nobr>159

Differential equations and their general solutions, as well as nonexistence theorems for six invariant classes of the vacuum constraint equations

S. E. Stepanov^ab, J. Mikeš^c

^a Financial University under the Government of the Russian Federation, Moscow, Russia
^b Russian Institute for Scientific and Technical Information of the~Russian Academy of Sciences, Moscow, Russia
^c Department of Algebra and Geometry, Palacky University, Olomouc, Czech Republic

Abstract: In this paper, we classify the vacuum constraint equations of General Relativity using the representation theory of classical groups. We also provide general solutions for the differential equations of each class and prove vanishing (nonexistence) theorems for these solutions using the Bochner technique, which is an important tool in Geometric Analysis.

Keywords: Riemannian manifold, pointwise orthogonal decomposition, general relativistic constraint equations.

MSC: MSC 2010: 53A45; 53C20

Received: 18.03.2025
Revised: 01.04.2025

DOI: 10.4213/tmf10985