Riemannian submersion

In the mathematical field of differential geometry , a Riemannian submersion is a submersion of one Riemannian manifold to another, which respects the Riemannian metric and which looks locally like an orthogonal projection onto the tangential space of the second manifold.

definition

Let and be two Riemannian manifolds and ${\displaystyle (M,g)}$ ${\displaystyle (N,h)}$

${\displaystyle f\colon M\to N}$

a submersion.

Then a Riemannian submersion is called if the isomorphism ${\displaystyle f}$

${\displaystyle df\colon \mathrm {ker} (df)^{\perp }\rightarrow TN}$

is an isometry .

Construction of metrics on quotient spaces

A Lie group works isometric, free and actually discontinuous on a Riemannian manifold . The quotient space is a differentiable manifold and one has an isomorphism . ${\displaystyle G}$ ${\displaystyle (M,g)}$ ${\displaystyle N=M/G}$ ${\displaystyle df\colon \mathrm {ker} (df)^{\perp }\rightarrow TN}$

A Riemannian metric is clearly defined by the condition that this isomorphism should be an isometry . It is known as the quotient metric. With this metric, the quotient mapping becomes a Riemannian submersion. ${\displaystyle N}$ ${\displaystyle f\colon M\to N}$

Examples

The Fubini Study metric on the complex projective space is the quotient metric for the standard effect of the circle group on the "round sphere", that is, the sphere of constant cutting curvature +1. The quotient mapping is with this metric ${\displaystyle \mathbb {C} P^{n}=S^{2n+1}/S^{1}}$

${\displaystyle f\colon S^{2n+1}\to \mathbb {C} P^{n}}$

thus a Riemannian submersion.

For this is the Hopf grain of the standard sphere : the Hopf map ${\displaystyle n=1}$ ${\displaystyle S^{3}}$

${\displaystyle H\colon S^{3}\to S^{2}}$

gives a Riemannian submersion.

O'Neill formula

The sectional curvature of the image space of a Riemannian submersion can be calculated from the sectional curvature of the original image space with the O'Neill formula :

${\displaystyle K_{N}(X,Y)=K_{M}({\tilde {X}},{\tilde {Y}})+{\tfrac {3}{4}}|[{\tilde {X}},{\tilde {Y}}]^{V}|^{2}}$.

Here orthonormal vector fields are on , their horizontal elevations are on , denotes the commutator of vector fields and is the projection of the vector field onto the vertical distribution . ${\displaystyle X,Y}$ ${\displaystyle N}$ ${\displaystyle {\tilde {X}},{\tilde {Y}}}$ ${\displaystyle M}$ ${\displaystyle [*,*]}$ ${\displaystyle Z^{V}}$ ${\displaystyle Z}$

literature

• Jeff Cheeger , David G. Ebin : Comparison theorems in Riemannian geometry. Revised reprint of the 1975 original. AMS Chelsea Publishing, Providence, RI, 2008. ISBN 978-0-8218-4417-5