Totally geodesic submanifold
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1 Definition
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:
![II(X,Y)= \overline{\nabla}_XY - \nabla_XY\,.](/images/math/c/9/f/c9f0202dd04bd6320bb420f566b05dc6.png)
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
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
![\left\{y \in \overline{M}; f(y)=y\right\}](/images/math/5/e/3/5e3ac81fd83948fcfc7757b1df09fce4.png)
with the induced Riemannian metric is a totally geodesic submanifold.
2 Examples
Example 2.1.
- 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 isometry
One 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].
References
- [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