Totally geodesic submanifold
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We consider a submanifold of a Riemannian manifold . The Riemannian metric induces a Riemannian metric on the submanifold Then is also called a Riemannian submanifold of the Riemannian manifold
Definition 1.1. A submanifold of a Riemannian manifold is called totally geodesic if any geodesic on the submanifold with its induced Riemannian metric is also a geodesic on the Riemannian manifold
General references are [Chen2000, ch.11] and [Helgason1978, I §14]. On the Riemannian manifold resp. there exists an unique torsion free and metric connection resp. It is called the Levi-Civita connection. Then the shape tensor or second fundamental form tensor is a symmetric tensor field which can be defined as follows for tangent vectors resp. vector fields on the submanifold:
Proposition 1.2 (cf. [O'Neill1983, p.104]). For a Riemannian submanifold of the Riemannian manifold the following statements are equivalent:
- is a totally geodesic submanifold of
- The shape tensor vanishes:
- For a vector tangential to the submanifold the geodesic on the Riemannian manifold defined on a small interval with initial direction stays on the submanifold.
Part (c) implies that locally a totally geodesic submanifold is uniquely determined by the vector subspace for some , provided that is connected and complete. There is a result by É. Cartan providing necessary and sufficient conditions for the existence of a totally geodesic submanifold tangential to a given vector subspace of the tangent space in terms of the curvature tensor, cf. [Chen2000, 11.1]. This result shows that for most Riemannian manifolds no totally geodesic submanifolds of dimension at least two exist. On the other hand totally geodesic submanifolds do occur if the manifold carries isometries:
Theorem 1.3 (cf. [Klingenberg1995, 1.10.15]). Let be an isometry of the Riemannian manifold Then every connected component of the fixed point set
with the induced Riemannian metric is a totally geodesic submanifold.
- A geodesic can be viewed as a totally geodesic submanifold of dimension one.
- Consider the standard sphere For the -sphere
is a totally geodesic submanifold of It is the fixed point set of the isometryOne can see immediately that any complete -dimensional totally geodesic submanifold of is of this form up to an isometry of the sphere.
- We denote by the -dimensional complex projective space of one-dimensional linear subspaces of the complex vector space For the inclusion induces an inclusion of the -dimensional complex projective space into This is a totally geodesic submanifold since it is the fixed point set of the isometry on induced by the reflection
The last two examples are in particular examples of symmetric spaces. The totally geodesic submanifolds of a symmetric space can be described in terms of a Lie triple system, cf. [Helgason1978, ch.IV, §7] or [Chen2000, 11.2].
- [Chen2000] B. Chen, Riemannian submanifolds, in Hanbook of differential geometry, Vol. I, edited by F. J. E. Dillen and L. C. A. Verstraelen, North-Holland, 2000, 187–418. MR1736854 (2001b:53064) Zbl 1214.53014
- [Helgason1978] S. Helgason, Differential geometry, Lie groups, and symmetric spaces, Academic Press, Orlando, Dan Diego New York 1978. MR514561 (80k:53081) Zbl 0993.53002
- [Klingenberg1995] W. P. A. Klingenberg, Riemannian geometry, Walter de Gruyter & Co., 1995. MR1330918 (95m:53003) Zbl 1073.53006
- [O'Neill1983] B. O'Neill, Semi-Riemannian geometry, Academic Press Inc., 1983. MR719023 (85f:53002) Zbl 0531.53051