Let $(M,g)$ be a Riemannian manifold, and $p,q\in M$ be two fixed points. We assume $p,q$ are close enough. Say, we assume $p$ and $q$ are in the same normal coordinate chart. It is clear that there is some diffeomorphism $F:M\to M$ so that $F(p)=q$ and $F$ equals the identity outside a small neighborhood of $p$ and $q$. For example, we take $F=\exp(X)$ for some vector field $X$ which flows from $p$ to $q$ and which is supported near $p$ and $q$.



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> **Question**: Now, suppose $F'$ is another such diffeomorphism ($F'(p)=q$ and equal to the identity outside a small neighborhood of $p$ and $q$ too). Can we find an isotopy of diffeomorphisms $F\_s,s\in [0,1]$ so that $F\_0=F$, $F\_1=F'$ and for each $s$ we have $F\_s(p)=q$? Moreover, can we require $F\_s=\exp(X\_s)$ for a smooth family of vector fields $X\_s,s\in[0,1]$?
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I guess the answer should be positive. For example, in the special case $p=q$ and $F'=\mathrm{id}$; write $F=\exp(X)$ and then $X\_p=0$. Thus, $F\_s=\exp(sX)$ suffices.
My attempt in general is that if $F'=\exp(X')$ then we may first take $G\_s=\exp((1-s)X+sX')$. But we cannot guarantee $G\_s(p)=q$. Maybe we can do some modification.



Existence of an isotopy in Riemannian manifold
For the first question, which concerns just the smooth category without reference to metrics, the answer depends on the dimension of $M$. Before explaining this a small clarification is needed. By "a small neighborhood of $p$ and $q$" you probably mean a ball containing $p$ and $q$, otherwise one could take the neighborhood to consist of disjoint balls about $p$ and $q$, and then there would be no chance of finding a diffeomorphism $F$ with $F(p)=q$ and with $F$ the identity outside the neighborhood.


Let $Diff(D^n)$ be the group of diffeomorphisms $F:D^n\to D^n$ which are the identity in a small neighborhood $N$ of the boundary of $D^n$, with the $C^\infty$ topology on $Diff(D^n)$. The set $\pi\_0Diff(D^n)$ of path components of $Diff(D^n)$ is the same as the set of isotopy classes of diffeomorphisms in $Diff(D^n)$ with the understanding that isotopies $F\_s$ restrict to the identity on $N$ for all $s$. You are interested in the subspace $Diff(D^n;p,q)$ consisting of diffeomorphisms taking $p$ to $q$. It is not hard to see that every $F\in Diff(D^n)$ can be isotoped to take $p$ to $q$, and also that any isotopy $F\_s$ between diffeomorphisms $F\_0,F\_1$ taking $p$ to $q$ can be deformed, fixing $F\_0$ and $F\_1$, to an isotopy with $F\_s(p)=q$ for all $s$. In other words the natural map $\pi\_0Diff(D^n;p,q)\to \pi\_0Diff(D^n)$ is a bijection. (In fact this holds for all higher homotopy groups as well.)


If $\pi\_0Diff(D^n)=0$, i.e., there is a single isotopy class, then the the answer to the first question is Yes. It is known that $\pi\_0Diff(D^n)=0$ for $n=1$ (elementary), $n=2$ (Smale), and $n=3$ (Cerf). For $n=4$ it is still unknown whether $\pi\_0Diff(D^n)$ is $0$ or not. For $n\geq 5$ it is known that $\pi\_0Diff(D^n)$ is isomorphic to the group of exotic spheres of dimension $n+1$ (via the h-cobordism theorem and Cerf's pseudoisotopy theorem). This group is known to be $0$ for $n=5,11,55,60$ and it is known to be nonzero for all other $n\leq 125$ and all other even $n$, using hard calculations in the stable homotopy groups of spheres.


An embedding $D^n \subset M$ induces maps $\pi\_0Diff(D^n)\to\pi\_0Diff(M)$ and $\pi\_0Diff(D^n;p,q)\to\pi\_0Diff(M;p,q)$ by extending diffeomorphsms via the identity outside $D^n$. In these terms the question becomes whether the image of the map $\pi\_0Diff(D^n;p,q)\to\pi\_0Diff(M;p,q)$ is $0$. If $M$ is simply connected then it is not hard to show that the map $\pi\_0Diff(M;p,q)\to \pi\_0Diff(M)$ is a bijection, so in these cases the question boils down to whether the image of the map $\pi\_0Diff(D^n)\to\pi\_0Diff(M)$ is $0$. The answer is certainly Yes if $\pi\_0Diff(D^n)=0$, so the answer is Yes for $M$ simply connected of dimension $n=1,2,3,5,11,55,60$. In the special case that $M$ is the sphere $S^n$ there is a classical elementary argument showing that the map $\pi\_0Diff(D^n)\to\pi\_0Diff(S^n)$ is injective (also on all higher homotopy groups), so this provides a No answer for $M=S^n$ for all other $n\leq 125$ and all even $n\geq 126$. For other manifolds $M$ the question of injectivity of $\pi\_0Diff(D^n)\to\pi\_0Diff(M)$ was studied in the heyday of differential topology but I can't recall any specific results or references at the moment.


