# An intrinsic parallel transport in Wasserstein space

## TL;DR

This paper introduces a geometric method for parallel transporting tangent cones along geodesics in Wasserstein space over a Riemannian manifold, aligning with previous formal calculations in smooth cases.

## Contribution

It provides a rigorous geometric construction of parallel transport in Wasserstein space, extending formal calculations to a precise mathematical framework.

## Key findings

- Constructs a geometric parallel transport in Wasserstein space
- Shows agreement with formal calculations in smooth cases
- Extends understanding of geometric structures in optimal transport

## Abstract

If M is a smooth compact connected Riemannian manifold, let P(M) denote the Wasserstein space of probability measures on M. We describe a geometric construction of parallel transport of some tangent cones along geodesics in P(M). We show that when everything is smooth, the geometric parallel transport agrees with earlier formal calculations.