Riemannian Geometry Based on the Takagi's Factorization of the Metric Tensor

TL;DR

This paper introduces a novel approach to Riemannian geometry using Takagi's factorization of the metric tensor, offering new insights into curvature and manifold properties relevant to physics.

Contribution

It proposes a new analytical framework for Riemannian geometry based on Takagi's factorization, differing from traditional metric and connection-based methods.

Findings

Provides conditions to distinguish curved and flat manifolds.

Offers a new decomposition of the curvature tensor into canonical parts.

Introduces intermediate geometric objects for geometric analysis.

Abstract

The Riemannian geometry is one of the main theoretical pieces in Modern Mathematics and Physics. The study of Riemann Geometry in the relevant literature is performed by using a well defined analytical path. Usually it starts from the concept of metric as the primary concept and by using the connections as an intermediate geometric object, it is achieved the curvature and its properties. This paper presents a different analytical path to analyze the Riemannian geometry. It is based on a set of intermediate geometric objects obtained from the Takagi's factorization of the metric tensor. These intermediate objects allow a new viewpoint for the analysis of the geometry, provide conditions for the curved vs. flat manifolds, and also provide a new decomposition of the curvature tensor in canonical parts, which can be useful for Theoretical Physics.