Riemannian product

In the mathematical field of differential geometry , the Riemann product is the product of two Riemannian manifolds with the product metric. ${\displaystyle M\times N}$

The 2-dimensional torus as the product of two circles.

definition

Are and two Riemannian manifolds and their Cartesian product with the product topology and the projections and on the two factors, so defined ${\displaystyle (M,g)}$ ${\displaystyle (N,h)}$ ${\displaystyle M\times N}$ ${\displaystyle p\colon M\times N\to M}$ ${\displaystyle q\colon M\times N\to N}$

${\displaystyle (g\oplus h)_{m,n}(u,v)=g_{m}(Dp(u),Dp(v))+h_{n}(Dq(u),Dq(v))}$

for a Riemannian metric on . The manifold with the Riemannian metric is called the Riemannian product of and . ${\displaystyle m\in M,n\in N,u,v\in T_{(m,n)}(M\times N)}$ ${\displaystyle g\oplus h}$ ${\displaystyle M\times N}$ ${\displaystyle M\times N}$ ${\displaystyle g\oplus h}$ ${\displaystyle (M,g)}$ ${\displaystyle (N,h)}$

Examples

The product of two circles is a torus with a flat metric. More generally, in every Riemannian product there are planes of section curvature 0: If a geodesic is in and a geodesic is in, then there is a flat submanifold of . ${\displaystyle k}$ ${\displaystyle M}$ ${\displaystyle l}$ ${\displaystyle N}$ ${\displaystyle k\times l}$ ${\displaystyle M\times N}$

literature

• W. Klingenberg : Riemannian Geometry , de Gruyter 1982; Section 1.8