# Banach manifold of paths with endpoints on submanifolds

Fix a Riemannian metric on a manifold $M$. Suppose that we fix two points $x,y \in M$. We start with the space

$C^{\infty}_{\searrow}(x,y) = \left\{\gamma: \mathbb{R}\to M:\,\lim_{t\to-\infty}\gamma(t) = x, \lim_{t\to\infty}\gamma(t)=y, \exists\,C,\delta > 0 \text{ such that} \vert\gamma'(t)\vert \leq Ce^{-\delta\vert t\vert}\right\}$

We can complete this space to a Banach manifold, $\mathcal{P}^{1,p}(x,y)$. The charts are given by pairs $(U_{\gamma},\gamma^{*}\exp)$, where $\gamma \in C^{\infty}_{\searrow}(x,y)$, $\exp: TM \to M$ is the exponential map coming from our metric, and $U_{\gamma} \subseteq W^{1,p}(\gamma^{*}TM)$ is chosen small enough so that for any $X \in U_{\gamma}$, $\exp_{\gamma(t)}{X(t)}$ stays within an injectivity radius of $\gamma(t)$.

I'm interested in a generalization of this construction when the endpoints of my curves lie in submanifolds, $S_{1},S_{2}\subseteq M$. I could try and do the same thing as above, varying over all pairs $(x,y) \in S_{1}\times S_{2}$:

$\mathcal{P}^{1,p}(S_1,S_2) := \coprod_{(x,y) \in S_1\times S_2} \mathcal{P}^{1,p}(x,y)$

But this gives me a space with the "wrong" topology. For example, any two paths with different endpoints are not in the same path component of $\mathcal{P}^{1,p}(S_1,S_2)$ as I've defined it here. I want paths with nearby endpoints to be considered close together.

To fix the previous problem, I could instead complete to a space where my local model is:

$C^{k}_{S_1,S_2}(\gamma^{*}TM) = \left\{X \in C^{k}(\gamma^{*}TM):\, \lim_{t\to-\infty} X(t) \in TS_1, \lim_{t\to\infty} X(t) \in TS_2\right\}$

This local model allows for variations of the end points; however, the drawback is that I can't use the same charts as I did before. The problem is that $S_1$ and $S_2$ might not be totally geodesic manifolds, so my endpoints will detach from $S_1$ and $S_2$ when I try and deform through some $X \in C^{k}_{S_1,S_2}(\gamma^{*}TM)$ using the exponential map. For the applications I have in mind, assuming that $S_1$ and $S_2$ are totally geodesic submanifolds is too restrictive.

Here are my questions:

1. Is there already in the literature a construction of a Banach manifold of paths whose endpoints are on fixed submanifolds? If it helps, I'm interested in this space in the context of Morse-Bott theory.

2. Is there any other construction which allows me to produce a 1-parameter family of curves $\gamma_s(t)$ such that $\partial_s\gamma = X$, and so that the endpoints of $\gamma_s$ are always on $S_1$ and $S_2$?

• I think that your boundary conditions at infinity together with the growth assumption for the derivative lets you reparametrize the paths with $[0, 1] \ni s \mapsto \frac{1}{\delta}(\log(s)-\log(1-s)) \in (-\infty, \infty)$, which should be a diffeomorphism. Hence you can consider a compact interval, and then the answer below applies. Oct 26 '15 at 22:44
• Oh now I see that the $\delta$ need not be uniform over your space. This might be a problem for the approach. Oct 26 '15 at 22:47

For each point $s$ in a submanifold $S\subset M$, there exists some Riemannian metric on $M$, which makes a neighborhood $U\subset S$ of $s\in S$ totally geodesic in $M$. The definition of (open sets in) these mapping spaces is local in $S_1,S_2$ (see below), so the existence of metrics which make $S_\pm$ near given points totally geodesic is sufficient for the construction of these mapping spaces.

The auxiliary Riemannian metrics used here have nothing to do with other Riemannian metrics on M, which you may choose later for other purposes. (However these auxiliary metrics are all equivalent to any fixed Riemannian metric, so in the end, the exponential decay can still be specified by any fixed Riemannian metric on $M$.)

Now to the local model: Denote $S_-:=S_1,S_+:=S_2$. For a smooth curve $\gamma:\mathbb{R}\rightarrow M$ with $lim_{t\rightarrow \pm \infty} \gamma(t)=s_\pm$ and (small) open subsets $U_\pm\subset S_\pm$ with $s_\pm\in U_\pm$, one can consider the local model

$C^k_{U_-,U_+}(\gamma^*TM)=\{X\in C^k(\gamma^*TM) |\lim_{t\rightarrow \pm\infty}X(t)\in T_{s_\pm}S_\pm, exp^{U_\pm\subset M}_{s_\pm}(X(\pm \infty))\in U_\pm\}$

Here the exponential map is associated to some Riemannian metrics on tubular neighborhoods of $U_\pm\subset M$, which make $U_\pm\subset M$ totally geodesic.

In analogy to your space $C^\infty_{\searrow}(x,y)$, one may then consider variations of this space with some exponential decay condition on the component of $X$ normal to $S_\pm\subset M$ (in a tubular neighborhood of $S_\pm$, any vectorfield splits into two parts which are 'tangential resp. normal' to $S_\pm$).

• I understand your answer to mean: Choose metrics for the purposes of defining charts on my space, even though they might have nothing to do with the metric I fix from the start (which I am using to do other things, like define gradient flows of functions). I will have to think about whether or not this is compatible with the constructions that come later. Is the collection of metrics, for which a fixed embedded submanifold is totally geodesic, generic? Oct 28 '15 at 16:46
• 1.Yes, I mean this: To give the space of 'all' maps between manifolds (in your case with 'asymptotic boundary conditions') a Banach manifold structure, one can rely on quite explicit constructions, without relying on IFT or the existence of regular values. One can then use the IFT to to show that the zero set of a section of some bundle is a submanifold, (consisting of maps which solve a particular PDE), if the section is suitably generic/transverse. But the genericity is here only used in the second step, where e.g. the generic choice of some metric may make a certain section generic. Oct 28 '15 at 18:14
• 2. If you ask whether they form a countable intersection of open dense subsets for e.g. C^1 topology then no, they are not even dense; recall that the totally geodesic condition means that certain derivatives of the metric on the submanifold vanish. On the other hand it is easy to construct them locally, you can choose any chart for the submfld, which identifies it loc. with a linear subspace of R^n, and pull back the Euclidean metric to obtain loc. such a metric. However the claim is, that the local existence is sufficient for the constr. of the mapping space; then you can forget about them. Oct 28 '15 at 18:45
• After thinking about it for a while, you've convinced me that the only place where this choice of metric gets used is in getting charts (and thus also local coordinate expressions for sections), but the overall these choices don't affect any other constructions. Thanks for pointing this out! I've accepted your answer. Oct 29 '15 at 17:29

This is a bit long for a comment, and definitely not an answer. It is usually easier to work with paths coming from a compact interval. Then it is classical (see for example Klingenberg's books) to construct a Hilbert manifold structure on $H^1(I,M)$. The claim is now that the map $$P:H^1(I,M)\rightarrow M\times M$$ Which maps a path $c$ to its endpoints $(c(0),c(1))$ is submersive. Then for any submanifold $V$ of $M\times M$, the preimage $P^{-1}(V)$ is a submanifold of $H^{1}(I,M)$. If you take $V=S_1\times S_2$ you get a space of paths from $S_1$ to $S_2$. I don't think this nicely generalizes to a domain that is non-compact. A reference for such things that I have not read is "A convenient setting from global analysis" http://www.mat.univie.ac.at/~michor/apbookh-ams.pdf.

Another comment: Since you are interested in the Banach manifold structure (but not necessarily a choice of Finsler norm), can't you take a different metric for which the submanifolds are totally geodesic?

• Thank you for the reference. I'm working with $\mathbb{R}$ instead of a compact interval because I'm ultimately going to be interested in parameterized solutions to gradient flow equations. I have some freedom in my choice of metric, but I'm constrained in that I need Morse-Smale transversality for the descending and ascending manifolds. I wasn't able to convince myself that the collection of metrics satisfying Morse-Smale transversality, and such that a finite set of submanifolds are totally geodesic, is generic. Oct 26 '15 at 21:43
• For moduli spaces in Morse theory (i.e. not Floer theory) I find it usually easier to think of them in terms of intersections of stable and unstable manifolds. See for example Weber's expository paper or Austin and Braam's paper on Morse and Morse-Bott homology respectively. Oct 26 '15 at 22:02
• I've read the Austin-Braam paper and while it is definitely easier, I'm not sure if I can accomplish what I want using their ideas. I'm trying to build gluing maps for broken flow trajectories in Morse-Bott theory, in the presence of a group action. I'm trying to imitate the approach taken when your fixed points are isolated (as in Schwarz' book, or Audin-Damien), but I'm running into the problems that I've outlined in my question. Oct 26 '15 at 22:17
• Ok, that is clear! When I think of something better to say I'll return to the question. Oct 26 '15 at 22:22
• Thank you! I took a look at the book by Klingenberg, as you suggested, and the space that I need to be a Banach manifold definitely is one, and it even has the tangent space I expect it to have! I just need to figure out some charts for this thing that I can work with. Oct 26 '15 at 22:26