By Berndt, Jürgen; Console, Sergio; Olmos, Carlos Enrique
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Extra resources for Submanifolds and holonomy
Equivalently, for any curve c : [a, b] → M we have τc⊥ (Nc(a) ) = Nc(b) , where τc⊥ is the ∇⊥ -parallel transport along c. Recall that the rank of N is the dimension of any of its fibers. The following criterion is very useful in this context of submanifold theory in space forms (see [97, Chapter 4], or ). 1 (Reduction of codimension) Let f : M → M¯ be an isometric immersion from an m-dimensional connected Riemannian manifold M into an n¯ Assume that there exists a parallel subbundle dimensional standard space form M.
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.
This gives an explicit classification of the one-dimensional totally geodesic submanifolds of M¯ n (κ ). From this we also easily see that the canonical embeddings M¯ k (κ ) ⊂ M¯ n (κ ), 1 < k < n, are totally geodesic. The isometry group of M¯ n (κ ) acts transitively on the pairs (p,V ) with p ∈ M¯ n (κ ) and V a k-dimensional linear subspace of Tp M¯ n (κ ). 1 Let p ∈ M¯ n (κ ) and V be a k-dimensional linear subspace of Tp M¯ n (κ ), 0 < k < n. Then there exists a connected, complete, totally geodesic submanifold M of M¯ n (κ ) with p ∈ M and Tp M = V .
Submanifolds and holonomy by Berndt, Jürgen; Console, Sergio; Olmos, Carlos Enrique