Examples of Banach manifolds with function spaces as tangent spaces

Question

I have recently been learning the theory of Banach manifolds through Serge Lang's book on Differential Manifolds. So far the objects seem rather interesting but my intuition always comes from the finite dimensional case, which admittedly is very helpful but I certainly want to add onto this.

I am wondering if there is a reference where one has examples of Banach manifolds where the tangent spaces are function spaces (ie Sobolev spaces). Obviously, I would like to avoid the trivial example of the function space being the manifold.

In some sense I would like an example that embraces the infinite dimensional tangent space as well as giving me some understanding as to what the topological space/smooth structure should be in the setting.

Answer 1

This doesn't exactly fit your criteria as it's not a Banach manifold, but one good example of an infinite dimensional space is the space of probability measures with finite second moment along with the formal Riemannian metric induced by Otto Calculus.

You have to be careful with the functional analysis to make this rigorous, since the space is actually fairly singular. However, this is a really important construction since the inner product induces the 2-Wasserstein metric as its distance function, so provides a bridge between optimal transport and functional analysis. Just to give one example of an insight this provides, if you use this metric the heat equation is the gradient flow of the entropy.

Later edit: There have been a few questions in the comments so I figured I would expand on this answer.

In [1], Felix Otto constructs this space as well as its formal inner product. In this paper, most of proofs don't use this formalism, but Otto calculus has been developed quite a bit since then so it is possible to prove results using this approach.

For simplicity, we consider probability measures which are supported on a domain $\Omega \subset \mathbb{R}^n$ and restrict our attention to measures with finite second moment and which are absolutely continuous with respect to the Lebesgue measure. Doing so, we can consider the space of all the probability density functions $\rho$ as an infinite dimensional manifold.

There are many ways to think of the tangent space of this space, but the most natural way to do so is to consider signed measures $s$ with finite second moment and zero total mass. To see why this can be understood as the tangent space, we consider the ``nearby" probability measure $\rho_\epsilon = \rho + \epsilon s$. The integral of $\rho_{\epsilon}$ will be 1 and by restricting $s$ and $\rho$ to reside in a suitable function space (e.g., $C_c^\infty(\Omega)$) and to satisfy $\textrm{supp} (s) \subset \textrm{supp}(\rho)$, we can ensure that for $\epsilon$ sufficiently small, $\rho + \epsilon s$ is non-negative. So $$ s(x) = \lim_{\epsilon \rightarrow 0} \frac{\rho_\epsilon(x) - \rho(x)}{\epsilon}. $$

One of Otto's key insights is that it is possible to "change coordinates" for the tangent bundle by solving a particular elliptic differential equation. More precisely, we can consider a function $p$ which satisfies $$-\mathsf{div}(\rho \nabla p)=s, $$ which gives another way to understand the tangent space.

Doing so, there is a natural inner product that can be defined, which is $$ g_{\rho}\left(s_{1}, s_{2}\right)=\int_{X} (\nabla p_1 \cdot \nabla p_{2}) \rho dx $$

If one uses this as a metric, the associated distance turns out to be the Wasserstein distance, and there are many analytic results which follow from this formulation. However, care must be taken since this calculation is only formal. It is also worth noting that this is distinct from the Fisher metric, which is another Riemannian metric on spaces of probability measures. For metrics which interpolate between these two, see the following papers [2-4].

[1] Otto, Felix, The geometry of dissipative evolution equations: The porous medium equation, Commun. Partial Differ. Equations 26, No. 1-2, 101-174 (2001). ZBL0984.35089.

[2] Chizat, L., Peyré, G., Schmitzer, B. and Vialard, F.X., 2018. An interpolating distance between optimal transport and Fisher–Rao metrics. Foundations of Computational Mathematics, 18, pp.1-44.

[3] Kondratyev, S., Monsaingeon, L. and Vorotnikov, D., 2016. A new optimal transport distance on the space of finite Radon measures.

[4] Liero, M., Mielke, A. and Savaré, G., 2018. Optimal entropy-transport problems and a new Hellinger–Kantorovich distance between positive measures. Inventiones mathematicae, 211(3), pp.969-1117.

Answer 2

Banach manifolds as a natural generalization of the finite dimensional case became “popular” in the late 60s. In Physics an obvious example would be the projective Hilbert space, the state space. Klingenberg’s book on Riemannian Geometry has some chapters on (Hilbert) Riemannian Geometry. Infinite Hamiltonian systems and symplectic geometryare treated extensively by many authors (E.G. Marsden, Ratiu, Weinstein etc.).

However results by Eells/Elworthy 1970, Elworthy 1972, Henderson 1969, 1972, show that Hilbert manifolds are essentially open subsets of their tangent space and smooth Banach manifolds are more or less all diffeomorphic (Henderson 1972). Also it would appear natural to look at the diffeomorphism group of a finite dimensional manifold. The tangent space should be given by the space of vector fields. Unfortunately a result by (Omori 1970) shows Banach manifolds are not suitable in that important case, more general (topological) vector spaces are needed. See the introduction of Kriegl/Michor 1997: A Convenient Setting of Global Analysis.