Torsion and the second fundamental form for distributions

TL;DR

This paper generalizes the second fundamental form to manifolds with linear connections and integrable distributions, linking it to torsion and the shape map, extending classical Riemannian concepts.

Contribution

It introduces a generalized second fundamental form for distributions, incorporating torsion and relating it to the shape map, broadening classical Riemannian geometry.

Findings

Defines a bilinear form for distributions with torsion

Connects the form to the shape map of the connection

Generalizes the shape operator to codimension one cases

Abstract

The second fundamental form of Riemannian geometry is generalised to the case of a manifold with a linear connection and an integrable distribution. This bilinear form is generally not symmetric and its skew part is the torsion. The form itself is closely related to the shape map of the connection. The codimension one case generalises the traditional shape operator of Riemannian geometry.