# Examples of Banach manifolds with function spaces as tangent spaces

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.

• Take $W^{p,k}(M, N)$ (embed $N$ in $\Bbb R^K$ for large $K$ to make sense of this; take p,k so that all maps are continuous). then the tangent space to f: M -> N ought to be the space of sections $W^{p,k}(f^*TN)$, if I remember right.
– mme
Feb 14, 2021 at 12:28
• If you are willing to consider the more general setting of Fréchet spaces, then the diffeomorphism group is an example of a Fréchet manifolds that is not a function space (but a subset of a function space) whose tangent spaces are function spaces. Feb 14, 2021 at 13:26
• @MikeMiller I am not quite sure I understand what you are referring but I think this in spirit of what I am search for. Can I get a reference? Feb 14, 2021 at 15:56
• I don't know a reference that spells out all details (instead of simply stating the result; if you want that you can find it in McDuff-Salamon "J-holomorphic curves and symplectic topology"). You get coordinate charts near smooth maps $f$ by sending a small section $\xi$ of $f^*TM$ to the flow $\text{exp}_{\xi} \circ f$. You might be able to find things with the keyword "Banach manifolds of Sobolev mappings" or something.
– mme
Feb 14, 2021 at 16:07
• In general, when $p,k$ are chosen so that we have the inclusion in the space of continuous functions, the space of sections $W^{p,k}(F)$ of a fibre bundle $\pi:F\to M$ over a closed manifold $M$ is a well-defined Banach manifold, and its tangent bundle is $W^{p,k}(VF)$, for $VF$ the vertical tangent bundle (i.e. $VF_x=\lbrace u \in TF, T\pi(u)=(x,0)\in T_xM\rbrace$). Feb 14, 2021 at 16:35

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 There are many ways to think of the tangent space. The natural coordinates are to consider all signed measures with finite second moment and zero total mass. However, one of Otto's key insights was that if you "change coordinates" by solving a particular elliptic differential equation (whose coefficients depend on the base point), there is a natural inner product that one can define

You have to be careful with the functional analysis to make this rigorous, and I believe the space is actually fairly singular (which is why it's merely formal). 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.

• Hi Gabe. This is rather interesting. In fact from a quick read of statistical manifolds I believe that probability measures do form a manifold? Can you refer me to a simple example of this 'change in coordinates'? Feb 14, 2021 at 15:51
• Felix Otto's paper "The geometry of dissipative evolution equations the porous medium equation" gives some excellent intuition at the beginning. I can't find a reference which states explicitly that the map $s \mapsto p$ where $p$ satisfies $-\nabla \cdot(\rho \nabla p)=s$ is a change of coordinates, but that's one geometric way to think about it. Feb 14, 2021 at 19:32
• As far as probability measures forming a manifold, this depends in an essential way on the choice of topology. If you consider probability measures which are absolutely continuous with respect to some reference measure and use something like the KL-divergence to induce the topology, then I believe this does induce a manifold structure (with a rather strong topology). On the other hand, the Wasserstein 2-distances this induces the weak topology and I believe (though can't find a reference offhand) that the space ends up being singular. If I find a reference, I'll edit it in. Feb 14, 2021 at 19:46

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.

• Nice answer. Another important example of tangent spaces not working as they would in a Banach manifold is the unitary group of a separable infinite dimensional Hilbert space, when given the strong operator topology. The tangent space at the identity ought to be the unbounded self-adjoint operators, but this unfortunately is not a vector space. I think the unitary group in the norm topology is a Banach manifold, the tangent space is the bounded self-adjoint operators, but unfortunately this is not the topology on the unitary group that we want to use in practice (e.g. quantum mechanics). Feb 21, 2021 at 4:49