Total geodetic submanifold

Total geodetic submanifolds occur in differential geometry , a branch of mathematics . They generalize the concept of hyperplanes in Euclidean spaces to Riemannian manifolds .

definition

A submanifold of a Riemannian manifold is called totally geodesic if every geodesic in is also a geodesic in . ${\ displaystyle N}$ ${\ displaystyle M}$ ${\ displaystyle N}$ ${\ displaystyle M}$

An equivalent condition is that the second fundamental form of is identical . ${\ displaystyle N}$ ${\ displaystyle 0}$

Examples

If an isometry is a Riemannian manifold, then the fixed point is -set ${\ displaystyle f \ colon M \ rightarrow M}$

{\ displaystyle N: = \ left \ {x \ in M: f (x) = x \ right \}}

a totally geodetic submanifold.

Planes in Euclidean are sets of fixed points of reflections and therefore totally geodetic surfaces. ${\ displaystyle \ mathbb {R} ^ {3}}$

More generally, subspaces of Euclidean are totally geodetic.

{\ displaystyle \ mathbb {R} ^ {n}}

Great circles on the sphere are also fixed point sets of reflections and therefore totally geodetic.

For the projective space is a totally geodetic submanifold of and a totally geodetic submanifold of . ${\ displaystyle k <n}$ ${\ displaystyle P ^ {k} \ mathbb {C}}$ ${\ displaystyle P ^ {n} \ mathbb {C}}$ ${\ displaystyle P ^ {k} \ mathbb {R}}$ ${\ displaystyle P ^ {n} \ mathbb {R}}$

Many Riemannian manifolds do not have any total geodetic submanifolds of codimension 1.

A surface in a hyperbolic -manifold is homotopic to a total geodetic surface if and only if it is acylindrical . ${\ displaystyle 3}$

The total geodetic surfaces of hyperbolic 3-manifolds form dense subsets in the Teichmüller spaces of closed, orientable surfaces.

literature

do Carmo, Manfredo Perdigão: Riemannian geometry. Translated from the second Portuguese edition by Francis Flaherty. Mathematics: Theory & Applications. Birkhauser Boston, Inc., Boston, MA, 1992. ISBN 0-8176-3490-8

Web links

Manifold Atlas

Individual evidence

↑ Jost, Jürgen: Riemannian geometry and geometric analysis. Sixth edition. University text. Springer, Heidelberg, 2011. ISBN 978-3-642-21297-0 (Theorem 3.4.3)
↑ Fujii, Michihiko; Soma, Teruhiko: Totally geodesic boundaries are dense in the moduli space. J. Math. Soc. Japan 49 (1997), no. 3, 589-601.

[1] Jost, Jürgen: Riemannian geometry and geometric analysis. Sixth edition. University text. Springer, Heidelberg, 2011. ISBN 978-3-642-21297-0 (Theorem 3.4.3)

[2] Fujii, Michihiko; Soma, Teruhiko: Totally geodesic boundaries are dense in the moduli space. J. Math. Soc. Japan 49 (1997), no. 3, 589-601.