The Schwarzian Curvature

TL;DR

This paper explores the Schwarzian curvature as a geometric interpretation of the Schwarzian derivative, extending its concepts to higher-dimensional projective spaces and providing detailed formulas and examples.

Contribution

It extends the understanding of Schwarzian curvature to higher dimensions and offers detailed formulas and properties, building on Gao's work.

Findings

Derived formulas for Schwarzian curvatures in CP, CP^2, and CP^3.

Analyzed transformation rules under coordinate changes.

Provided explicit examples of Schwarzian curvature calculations.

Abstract

We start with introducing one of the most fundamental notions of differential geometry, Manifolds. We present some properties and constructions such as submanifolds, tangent spaces and the tangent map. Then we continue with introducing the real and complex projective space, and describe them from some different points of view. This part is finished by showing that CP^n is a Grassmannian manifold. At this stage we are ready to present the main subject of this thesis. The Schwarzian curvature, usually seems to be an accidental by-product of the calculations, can be seen as a geometric interpretation of the Schwarzian derivative. Flanders interpreted the Schwarzian derivative of a C''' function as a curvature for curves in the projective line by using the moving frame method of Cartan. The same argumentation was extended by Gao to obtain the Schwarzian curvatures for curves in higher dimensional projective spaces. I give detailed presentation of Gao's work, where he presented the general formulas for the Schwarzian curvatures for curves in CP^n and gives some properties for the behaviour of the formulas, for example the transformation rules under change of coordinates. The Schwarzian curvatures for curves in CP, CP^2 and CP^3 are calculated, and some examples are given.