# The total intrinsic curvature of curves in Riemannian surfaces

## TL;DR

This paper investigates the intrinsic curvature of irregular curves on Riemannian surfaces, establishing a link between total curvature and an energy functional, and extending results via isometric embeddings.

## Contribution

It introduces a new energy functional for irregular curves' total intrinsic curvature and extends the analysis to Riemannian surfaces using isometric embeddings.

## Key findings

- Total intrinsic curvature equals a specific energy functional.
- Weak parallel transport is well-defined for curves with finite total curvature.
- Results extend to irregular curves in Riemannian surfaces via isometric embeddings.

## Abstract

We deal with irregular curves contained in smooth, closed, and compact surfaces. For curves with finite total intrinsic curvature, a weak notion of parallel transport of tangent vector fields is well-defined in the Sobolev setting. Also, the angle of the parallel transport is a function with bounded variation, and its total variation is equal to an energy functional that depends on the "tangential" component of the derivative of the tantrix of the curve. We show that the total intrinsic curvature of irregular curves agrees with such an energy functional. By exploiting isometric embeddings, the previous results are then extended to irregular curves contained in Riemannian surfaces. Finally, the relationship with the notion of displacement of a smooth curve is analyzed.