We have the following theorem in my lecture:
Let $M,N \subset P$ be Connected, totally geodesic and geodesically complete submanifolds, s.t. $\exists p \in M \cap N$ and $T_pM=T_pN$.
Then it holds $M=N$.
I understand this and also the proof for this. But then we have a corollary, that each connected, totally geodesic and geodesically complete submanifold of $\mathbb{R}^n_{\nu}$ is of the form $p + V$ where $V$ is a non degenerate subspace.
I see that the $p+V$ are connected, totally geodesic and geodesically complete submanifolds. But I don't see how it follows from the thm., that all .... submanifolds are of the form because we would have to show that for each other connected, totally geodesic and geodesically complete submanifold $M$ with $q \in M \cap (p+V)$ it holds $T_qN = T_q (p+V)$ and I don't see how this works.. can somebody give a hint?
Also, in my lecture it says that eacb connected totally geodesic submanifold of $\mathbb{R}^n_{\nu}$ is an open subset of a non degenerated affine subspace.. why should this be clear?
In order to show this corollary, one only need to observe that for any connected, totally geodesic and geodesically complete submanifold of $\mathbb{R}_\nu^n$, namely $N$ and for any $q\in N$, we have $T_q N = q + V$, where $V$ is some non-degenerate subspace of $\mathbb{R}_\nu^n$. And we have $T_q(q+V) \simeq q+V$. Then the corollary follows from the theorem.