Suppose $G\curvearrowright M$ is an isometric action of a Lie group on a complete Riemannian manifold $M$, and assume it is polar. This means that the action is proper and there exists a closed (hence complete) embedded submanifold $\Sigma\subseteq M$ (called a section) which meets all orbits orthogonally. It is well known that $\Sigma$ is totally geodesic, but I have not found a convincing proof of that fact.
There is a special case where the last statement is easy to prove: the second fundamental form vanishes at regular (and exceptional) points. Indeed, given any $p\in \Sigma$ such that $G\cdot p$ has maximal dimension, $v\in T_{p}(\Sigma)=T_{p}(G\cdot p)$ and $\xi\in T_{p}(G\cdot p)$, we can find an element $X$ in the Lie algebra $\mathfrak{g}$ of $G$ such that
$$\xi=X^{*}(p), \quad X^{*}(q)=\dfrac{d}{dt}\bigg|_{t=0}\operatorname{Exp}(tX)\cdot q.$$
The vector field $X^{*}$ is Killing, so that its covariant derivative is antisymmetric. Let $\mathbb{II}$ be the second fundamental form of $\Sigma$. Then $\mathbb{II}(v,v)$ is tangent to $G\cdot p$ and $\langle \mathbb{II}(v,v),\xi \rangle=-\langle v,\nabla_{v}X^{*} \rangle=0$, so $\mathbb{II}(v,v)=0$. Polarizing, we get $\mathbb{II}=0$.
The usual argument for proving that sections are totally geodesic at all points revolves around the fact that regular points are dense in $\Sigma$. My problem is that all proofs that I found seem to lack crucial details for it. Here are some examples:
- In "Lie Groups and Geometric Aspects of Isometric Actions", by Alexandrino and Bettiol, it is stated in Exercise 4.9 that the density follows from Kleiner's Lemma (cf Lemma 3.70), but I can't get the connection between the lemma and this fact.
- In "Critical Point Theory and Submanifold Geometry", by Palais and Terng, the authors state that density follows from the theory of Riemannian submersions, without giving further details.
- In "Polar Manifolds and Actions", by Grove and Ziller, the authors state that singular points are isolated along any geodesic, because of the Slice Theorem. This is because if $\gamma$ is any geodesic of $\Sigma$, and $t_{0}$ lies in the closure of $\{ t\in \mathbb{R}\colon \gamma(t)\in M_{R} \}$, then $\gamma(t_{0}-\varepsilon)$ and $\gamma(t_{0}+\varepsilon)$ have the same isotropy for sufficiently small $\varepsilon>0$ (again, because of the Slice Theorem), but I don't understand why this is the case.
Could somebody elaborate on any of the methods of proof proposed above (preferably the first or the last), or give a reference to a detailed proof of the fact that regular points are dense?
Thank you in advance!
EDIT
I forgot to mention another method of proof, proposed in the book "Submanifolds and Holonomy", by Berndt, Console and Olmos. I have an (almost full) solution, which needs to prove the following crucial fact: if $p\in \Sigma$ and there is an open subset $\Omega\subseteq \Sigma$ such that all orbits of the points in $\Omega$ have the same (nonprincipal) type, then $T_{p}(\Sigma)$ is pointwise fixed by the slice representation. If somebody could give a proof of this last fact, I would also accept it as an answer.
EDIT 2
I've decided to repost this question on MathOverflow. If anyone is interested in seeing it, you can find it on https://mathoverflow.net/questions/398008/sections-of-a-polar-action-are-totally-geodesic