A'global'Perspective on the Differential Geometry of Wasserstein Spaces
We develop a global differential calculus on the $L^2$-Wasserstein space over a closed Riemannian manifold, based on derivations of cylinder functions rather than on the standard pointwise approach. Within this framework we define some fundamental geometric tools. In particular, we show that the Levi-Civita connection on the base manifold lifts to the unique torsion-free connection compatible with the'extended'Otto metric. The corresponding Riemann tensor is exactly the lift of the base Riemann tensor, which shows that - in this framework - the correction terms of the classical gradient formalism are not intrinsic curvature terms, but arise from the projection onto the measure-dependent gradient distribution. This allows us to revisit with a global and purely differential approach the smooth computations by J. Lott, Comm. Math. Phys., 277(2):423-437, 2007, reaching partially different conclusions.