Cohomology for linearized Ricci curvature

Abstract: I will present full solvability and uniqueness conditions for the linearized Ricci curvature equations on compact Riemannian manifolds with boundary. These equations have long resisted analysis without restrictive curvature assumptions; the results I shall present are the first obtained without such restrictions in the interior. The approach relies on a generalized Hodge theory, based on microlocal methods, to reduce the problem to the study of two newly identified cohomologies. By means of Bochner technique, I also prove vanishing theorems for these cohomologies: notably, by incorporating connections on Riemannian metrics, the first cohomology vanishes if the boundary is convex, and the second cohomology vanishes if adapted to traceless-transverse (TT) tensors, and the boundary is round. I shall also discuss how this cohomological formulation captures the natural gauge group of geometric inverse problems and relates to the injectivity of the Lichnerowicz Laplacians on TT tensors.

mathematical physicsalgebraic topologydifferential geometryrepresentation theorystatistics theory

Audience: researchers in the topic

( paper | slides | video )

Prague-Hradec Kralove seminar Cohomology in algebra, geometry, physics and statistics

Series comments: Virtual coffee starts on Zoom already 15 minutes before the seminar.