Extensions of submanifold theory to non-real settings, with applications

TL;DR

This thesis extends classical Riemannian submanifold theory to non-real settings, introducing Rinehart spaces and holomorphic submanifold concepts, thereby unifying various geometries and revealing new connections between pseudo-Riemannian spaces.

Contribution

It develops a unified framework for Riemannian geometry over different ground rings and explores holomorphic submanifold theory with applications to Wick-related spaces.

Findings

Unified Riemannian geometry over various rings

Holomorphic submanifold theory in complex settings

Connections between Wick-related pseudo-Riemannian spaces

Abstract

In this thesis, we study extensions of the theory of Riemannian submanifolds in two directions. First, we will show how Riemannian geometry and submanifold theory in particular, can be generalized using the notion of 'Rinehart spaces', and it will be demonstrated how the developed framework unifies some existing and new flavours of Riemannian geometry over different ground rings. In the second part of the thesis, we give a description of holomorphic Riemannian submanifold theory where complex numbers fully replace the role of the real numbers as ground field, and show how this can be applied to reveal direct connections between the submanifolds of so-called Wick-related pseudo-Riemannian spaces.