Let be a differential manifold and a submanifold of . Let be a Riemannian metric on turning into a Riemannian manifold. Then, since each tangent space to is a subspace ofa tangentspace to , we can restrict to the tangent space to at each point. Further, the restriction will again be a smoothly varying positive definite symmetric bilinear form, and thus acquires the structure of a Riemannian manifold.
equipped with this Riemannian manifold structure is termed a Riemannian submanifold.
Levi-Civita connection on submanifold
The Levi-Civita connection on is not the same as that on . However, it is almost the same, in the sense that if and denote the Levi-Civita connections on and respectively, then for and vector fields on :
is, at each point, a vector normal to the submanifold in . Thus is the component of on the submanifold.
Metric space structure to the submanifold
A Riemannian manifold naturally has the structure of a metric space: the distance between two points being the infimum of the lengths of paths between them. This metric space structure is particularly useful for a complete Riemannian manifold, which is thus a geodesic metric space. A submanifold of a Riemannian manifold can thus be given two metric space structures:
- One, the metric space structure obtained by restricting the metric from the bigger manifold
- Two, the metric space structure obtained from the Riemannian metric on the submanifold
Distances in the second metric space structure are in general greater, because there may be paths in the biger manifold shorter than the shortest path in the smaller manifold. The two metric space structures coincide only for a totally geodesic submanifold.