Let $\alpha: [a,b]\to M$ be an embedded curve in a Riemannian manifold $(M,g)$ and let $p$ be a point in $M$, not on the curve $\alpha$. If $p$ is close enough to $\alpha$, there exists a unique geodesic $\gamma$ of $M$, parametrised proportional to arc length and satisfying $\gamma(0)= \alpha(s)$ for some $s \in [a,b]$, $g(\gamma'(0), \alpha'(s))= 0$ and $p = \gamma(1)$. The image of $p$ under the reflection with respect to $\alpha$ is the point $q = \gamma(-1)$. The $q$ said to be local reflection of $p$.
Now, if we take a non-zero vector field $X$ and use the above algorithm for integral curve of $X$ and suppose local reflection is a isometry, then what we can say about:
Q1: Riemann (or sectional) curvature of the manifolds? Is it completely determined?
Q2: the vector field $X$?
In some Papers
Take $p \in M$ and denote by $f$ the local reflection with respect to the integral curve of $X$ through $p$. Then $f(p) = p$,$f_∗X_p = X_p$ and $f_∗Y_p = -Y_p$ for $Y\in X_p^\perp$ . Since $f$ is a local isometry, it preserves the Riemann curvature tensor. In particular, for $Y_1 ,Y_2 ,Y_3 ∈ T_p M$, it holds
$$f_∗R_p (Y_1 ,Y_2 )Y_3 = R_p (f_∗ Y_ 1 ,f_∗Y_2 )f_∗ Y_3,$$ So, if $Y_1 ,Y_2 ,Y_3$ are orthogonal to $X$, we have $$f_∗R_p (Y_1 ,Y_2 )Y_3 = -R_p (Y_ 1 ,Y_2 )Y_3.$$
In this way, we see that, for $Y,Z,W$ orthogonal to $X$ it holds $$R(Y,Z )W \perp X,\ R(Y,Z )X = 0,\ R(Y,X)Z\sim X,\ R(Y,X)X ⊥ X. $$
Update: I found in some paper the following criterion for a reflection with respect to a curve to be locally isometric:
Theorem: Let $(M,g)$ be a Riemannianm manifold and $\alpha : [a,b]\to M$ an embedded curve in $M$. If the reflection with respect to the curve $\alpha$ is an isometry, then the following hold:
- $\alpha$ is a geodesic;
- $g((\nabla^{2k}_{u\cdots u}R)(u, v)u, \alpha^\prime) = 0;$
- $g((\nabla^{2k+1}_{u\cdots u}R)(u, v)u, w) = 0;$
- $g((\nabla^{2k+1}_{u\cdots u}R)(u, \alpha^\prime)u, \alpha^\prime) = 0;$
for all vectors $u , v$ and $w$ orthogonal to $\alpha^\prime$ and $k = 0, 1, 2, \cdots$. Moreover, if $M$ is analytic, these conditions are also sufficient for the reflection to be an isometry.
Thanks.
