This is my first question on this platform, I appreciate any suggestions on how to improve my question.
why is the Wasserstein space no manifold and in which way is its structure somehow similar to a manifold?
$(\mathcal{P}_2(\mathbb{R}^d), W_2)$ is no manifold: Assuming that the Wasserstein space $(\mathcal{P}_2(\mathbb{R}^d), W_2)$ is a manifold of dimension n, then the tangent space $Tan_\mu\mathcal{P}_2(\mathbb{R}^d)=\overline{\{\lambda(T-Id): (Id,T)_\#\mu\in\Pi_{opt}(\mu,T_\#\mu),\, \lambda>0\}}^{L_2(\mu,\mathbb{R}^d)}$ [Def. 8.5.1. Ambrosio] to any measure from the Wasserstein space would be a n-dimensional linear space. I want to show that the tangent space for an absolutely continuous measure is a linear subspace of L^2(\mu) and infinite dimensional whereas for a dirac measure the tangent space can be identified with $\mathbb{R}^d$. This would show that the Wasserstein space is not a manifold. The part for the dirac measure is clear, but for the absolute continuous case I already struggle to show that for $\mu$ absolutely continuous the tangent space is a linear space: How does one show that $$(Id,T)_\#\mu\in\Pi_{opt}(\mu,T_\#\mu),\, (Id,\tilde{T})_\#\mu\in\Pi_{opt}(\mu,\tilde{T}_\#\mu)\Longrightarrow (Id,T+\tilde{T})_\#\mu\in\Pi_{opt}(\mu,(T+\tilde{T})_\#\mu)$$ By the Brenier Theorem I know that there a exists an optimal map $L$ which fulfils $(Id,L)_\#\mu\in\Pi_{opt}(\mu,(T+\tilde{T})_\#\mu)$. I tried to show that $L = T+\tilde{T}$ $\mu$-a.e. or by assuming that they are not equal to get a map which pushes $\mu$ to $T_\#\mu$ and has lower costs than $T$.
Is there an easy way to show that the Wasserstein space is not locally Euclidean? Or do you know any reference where I can find the proof for this?
Thanks a lot for any help
