Given a pseudo-Riemannian manifold $(M,g)$, a submanifold $X$ is called totally-geodesic if every geodesic in $X$ (with respect to the induced metric $g_{ind}$) is also a geodesic when regarded in $M$.
In $X$ we can consider the second fundamental form induced by the ambient, defined as follows
$$ II(X,Y)=\nabla_XY-{\nabla_{ind}}_YX $$being $\nabla$ and $\nabla_{ind}$ the corresponding Levi-Civita connection of $g$ and $g_{ind}$ respectively, and $X,Y\in T_pX$.
Being a totally-geodesic manifolds is equivalent to $II=0$. See this webpage for more info and references.
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Author of the notes: Antonio J. Pan-Collantes
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