How can one prove that every geodesics of a submanifold M are geodesics on the "bigger" manifold N? Intuitively it makes alot of sense but i can't work with the "mathematical tools" that i have.
My tools:
i) I know that the Levi-Civita connection will satisfy: $$\nabla^M_XY=(\nabla_\overline{X}^N\overline{Y})^T$$ Where $\overline{X},\overline{Y}\in\mathcal{X}(N)$ are smooth local extensions of $X, Y ∈ \mathcal{X} (M)$ (we assume that they exist), and $(\cdot)^T$ denotes the orthogonal projection onto the respective tangent space of M.
ii) We call M totally geodesic if:
$$\nabla^M_XY=\nabla_\overline{X}^N\overline{Y}$$
The problem is: Prove that each geodesic of M is also one of N.