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 $S\_2$?



Banach manifold of paths with endpoints on submanifolds
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 $S\_\pm$).


