ABSTRACT
In this chapter we present the basics of submanifold theory in spaces of constant curvature, or briefly, in space forms. In the literature there are mainly three different definitions for a submanifold of a Riemannianmanifold. LetM and M¯ be Riemannian manifolds. When we have an isometric immersion from M into M¯, we say that M is an immersed submanifold of M¯. WhenM is a subset of M¯ and the inclusionM ↪→ M¯ is an isometric immersion, thenM is said to be a submanifold of M¯. If, in addition, the inclusion is an embedding, thenM is said to be an embedded submanifold of M¯. Note that a submanifold is embedded if and only if its topology coincides with the induced topology from the ambient space. The immersion of a real line as a figure eight in a plane is an example of an immersed submanifold that is not a submanifold. A dense geodesic on a torus is an example of a submanifold that is not embedded. The local theories for these three kinds of submanifolds are the same; the only difference arises when dealing with global questions. Therefore, when we deal with local properties of submanifolds, we make no distinction and just say submanifold.
