Tangent space to infinite dimensional manifolds

Question

In finite dimensional geometry, there is a single invariant of a vector space - its dimension. This characterizes finite dimensional manifolds as being glued from Euclidean balls.

This situation is quite different in infinite dimensions. There are papers talking about "zoo" of function spaces, which suggests that there is a very large variety of spaces.

Q: What are natural examples of smooth infinite dimensional manifolds in which the type of the tangent space varies from point to point?

Remark: The book of Kriegel and Michor "Convenient setting for Global analysis" has a section on the tangent bundle, but it does not contain natural examples.

Answer 1

A good example (and probably the first one appearing in the "Calculus of Variation in the Large") is the one of manifold valued maps, whose tangent space can be identified with vector fields tangent to the image, or to be more precise with sections of the pull back bundle.

To be more concrete, given a Riemannian manifold $(M,g)$ one can for instance consider the manifold $$ \mathcal M=\Bigl\{ \gamma :\mathbb S^1 \to M: \int_{\mathbb S^1} g_{\gamma(t)}(\dot \gamma(t), \dot \gamma(t)) dt<+\infty\Bigr\} $$ The tangent space to $\mathcal M$ at a curve $\gamma$ is then $$ T_\gamma \mathcal M=\Bigl\{ V: \mathbb S^1 \to TM: V(t)\in T_{\gamma(t)} M \Bigr\}. $$ The above is actually an Hilbert manifold with scalar product given by $$ \langle V, W\rangle_{\gamma} =\int_{\mathbb S^1} g_{\gamma(t)}(V(t),W(t)) dt $$ See for instance the book of Moore "Introduction to Global Analysis".

Answer 2

Infinite dimensional manifolds are modeled locally on one given infinite dimensional space. So I've never seen what you are asking for.

How different local models for infinite dimensional manifolds arise in practice is due to smoothness. Multiple local models can occur in the same problem. Here is a prototypical example:

Let $(M,g)$ be a Riemannian manifold. We can consider spaces of free loops inside $M$. There are obvious choices:

$C^\infty(S^1,M)$, the space of smooth loops, or $W^{1,2}(S^1,M)$ the space of absolutely continuous loops which are differentiable almost everywhere with $L^2$ bounded derivative.

The first one is a Frechet manifold, the second a Hilbert manifold. What is kind of bad about the first one is that it is hard to talk about Riemannian structures on the infinite dimensional manifold. The tangent spaces do not admit an inner product inducing the topology!

What is bad about the second example, is that some natural looking constructions do not give what you want. To resolve this you need to study different models. Let me delve a bit deeper in this:

The tangent space to a loop $\gamma\in W^{1,2}(S^1,M)$ are all vectorfields along $\gamma$ that are of $W^{1,2}$ regularity.

Let $\gamma\in W^{1,2}(S^1,M)$. Then we can compute its derivative $\dot \gamma$. Morally this should define an element in $T_\gamma W^{1,2}(S^1,M)$ as it is a vectorfield along gamma, and the map $\gamma\mapsto \dot \gamma$ should define a vector field over $W^{1,2}(S^1,M)$. However, the regularity is wrong, $\dot \gamma$ has lower regularity than $W^{1,2}(S^1,M)$. You can resolve this by introducing a "tangent like" bundle over $W^{1,2}(S^1,M)$ where the vector fields along the loops are merely square integrable, not $W^{1,2}$. In this sense there are two different "tangent bundles" over $W^{1,2}(S^1,M)$.

So this is a situation where different models play a role. Abstractly $L^2$ and $W^{1,2}$ are isomorphic. But one can think of Banach space settings (e.g., for example if the domain is not $S^1$ but a higher dimensional manifold) where this is not the case anymore.

One last remark: The spaces $C^{\infty}(S^1,M)$ and $W^{1,2}(S^1,M)$ are homeomorphic as topological spaces. Their differences only become apperent as infinite dimensional manifolds.