Exterior Generalised Geometry

TL;DR

This paper extends tools from semi-Riemannian geometry to generalized geometry, developing new notions like generalized curvature and applying them to Einstein equations and submanifold theory.

Contribution

It introduces generalized curvature concepts, such as the generalized second fundamental form, and applies them to Einstein equations and hypersurface theory in generalized geometry.

Findings

Defined generalized exterior curvature and mean curvature.

Derived generalized Gauss-Codazzi equations.

Established constraint equations for generalized Einstein gravity.

Abstract

It is the aim of this paper to transfer to generalised geometry tools employed in the study of semi-Riemannian immersions, specializing at times to semi-Riemannian hypersurfaces. Given an exact Courant algebroid $E \to M$ and an immersion $\iota\colon N \hookrightarrow M$, there is a well-known construction of an exact Courant algebroid $\iota^! E \to N$, the pullback of $E$. This paper explains the pullback of generalised metrics and divergence operators. Assuming $N$ is a hypersurface, it develops the notion of generalised exterior curvature, introducing the generalised second fundamental form and the generalised mean curvature. Generalised versions of the Gau{\ss}-Codazzi equations are obtained. As an application, the constraint equations for the initial value formulation of the generalised Einstein equations are established in the formalism of generalised geometry. Further applications include a generalised geometry version of the fundamental theorem for hypersurfaces and the result that generalised K\"ahler and hyper-K\"ahler structures restrict to submanifolds compatible with the generalised almost complex structure. In particular, we characterise exact semi-Riemannian Courant algebroids which are flat with respect to the canonical generalised connection. These play the role of the ambient space in the fundamental theorem mentioned above.