Two points with “identical” local geometry on Riemann Manifolds

Question

Let $S$ and $T$ be Riemann Manifolds of dimension $n$ and let $s$ be a point in $S$ and $t$ be a point in $T$. Is there a way to say that the local geometry of $s$ is “identical” to the local geometry of $t$, perhaps about some symmetry?

Answer 1

I would always be careful with the word "identical" in mathematics. In this situation, what I would say is that the local geometries of $s$ and $t$ are isometric is there exist open subsets $U \subset S$ and $V \subset T$, and an isometry $f : U \to V$, such that $s \in U$, $t \in V$, and $f(s)=t$. In case you don't the definition, to say that $f$ is an isometry means that $f$ is a diffeomorphism, and for every $x \in U$ and $y=f(x) \in V$ the derivative map $D_x f : T_x U \to T_y V$ is a linear isomorphism that preserves the inner products on those spaces (given by the Riemannian metrics), i.e. for each $v,w \in T_x U$ we have $\langle v,w \rangle = \langle D_x(v),D_x(w) \rangle$.