I was going through Helgason's paper titled "Totally Geodesic Radon Transform on Constant Curvature Spaces". In this paper, the author defines two types of homogeneous spaces out of a Lie group $G$, and its subgroups $K$ and $H_0$ (or $H_p$). Particularly, he defines $X = H/K$ and $\Xi = G/H_0$. Citing Chern's paper, he defines the notion of incidence as follows:
Definition: We say that $x \in X$ and $\xi \in \Xi$ are incident if as cosets they have a point in common.
Now, in Helgason's work, he takes $X$ as either $\mathbb{R}^n, \mathbb{S}^n$ or $\mathbb{H}^n$, $G$ as the group of isometries, $K$ as the isotropy group of the "origin" $o \in X$, and $H_0$ as the isotropy group of a totally geodesic submanifold passing through $o$. Since the definition of incidence of points is abstract, it carries on here as well. However, now we can look at the elements of the same set in two different ways: Elements of $X$ are points as well as cosets of $K$, and elements of $\Xi$ are cosets of $H_0$ as well as submanifolds of $X$. Therefore, we can talk about the distance between a point $x \in X$ and a submanifold $\xi \in \Xi$, in the sense of the distance induced by the Riemannian metric.
Helgason goes on to also characterize $\Xi$ in a different manner. He first fixes a totally geodesic submanifold $\xi_p \in \Xi$ which is at a distance $p \geq 0$ from $o$, and then looks at the isotropy group $H_p$ of $\xi_p$. With this, we can see that $\Xi \cong G/H_p$.
In this paper, Helgason claims that $x \in X$ is incident to $\xi \in \Xi$ if and only if $d \left( x, \xi \right) = p$, when we identify $\Xi$ with $G/H_p$. Although he states that this claim is "readily seen" to be true, I am not able to even come up with an idea to begin the proof. Every time I think of moving in one direction, I get confused with what notion of elements of $X$ and $\Xi$ is to be considered. Clearly, both notions have to be considered in order to connect them as the author suggests.
Any insights on this are appreciated!
Edit: I am now able to prove one sided implication from the if and only if statement. I would present my proof here for verification, along with the issue that I face to prove the other side of the implication.
First, let us assume that $x \in X \cong G/K$ and $\xi \in \Xi \cong G/H_p$ be incident. Then, there is some $h \in gK \cap \gamma H_p$, where $x = gK$ and $\xi = \gamma H_p$. With this, we see that $g^{-1} h \in K$ and $\gamma^{-1} h \in H_p$. Also, all of them (group elements) are isometries on $X$. Thus, we have
$$d \left( x, \xi \right) = d \left( \gamma^{-1} g \cdot o, \xi_p \right) = d \left( h^{-1} \gamma \gamma^{-1} g \cdot o, h^{-1} \gamma \xi_p \right) = d \left( o, \xi_p \right) = p.$$
This proves that whenever $x \in X$ and $\xi \in \Xi$ are incident (and $\Xi \cong G/H_p$), we have $d \left( x, \xi \right) = p$.
Now, to prove the converse, we need to assume that $d \left( x, \xi \right) = p$, and then produce an element $h \in gK \cap \gamma H_p$. However, all I can do for now is to deduce that
$$p = d \left( x, \xi \right) = d \left( \gamma^{-1} g \cdot o, \xi_p \right).$$
However, from here I cannot deduce what is required. All I can see is that $\gamma^{-1} g$ "rotates" the manifold $X$ about the submanifold $\xi$. However, although this geometrical intuition seems to be true in $\mathbb{R}^3$ by taking $o$ to be the origin, and $\xi$ to be a straight line, I am unable to realise it in higher dimensions, let alone in arbitrary manifolds.
Right now, it seems that the use of normal neighbourhood for a totally geodesic submanifold should work. However, again, I lack formalism in completing this idea as well!
Any help with this is highly appreciated!