Riemannian submersion

In differential geometry, a branch of mathematics, a Riemannian submersion is a submersion from one Riemannian manifold to another that respects the metrics, meaning that it is an orthogonal projection on tangent spaces. In differential geometry, a branch of mathematics, a Riemannian submersion is a submersion from one Riemannian manifold to another that respects the metrics, meaning that it is an orthogonal projection on tangent spaces. Let (M, g) and (N, h) be two Riemannian manifolds and f : M → N {displaystyle f:M o N} a (surjective) submersion, i.e., a fibered manifold. The horizontal distribution k e r ( d f ) ⊥ {displaystyle mathrm {ker} (df)^{perp }} is a sub-bundle of the tangent bundle of T M {displaystyle TM} which depends both on the projection f {displaystyle f} and on the metric g {displaystyle g} . Then, f is called a Riemannian submersion if and only if the isomorphism d f : k e r ( d f ) ⊥ → T N {displaystyle df:mathrm {ker} (df)^{perp } ightarrow TN} is an isometry. An example of a Riemannian submersion arises when a Lie group G {displaystyle G} acts isometrically, freely and properly on a Riemannian manifold ( M , g ) {displaystyle (M,g)} . The projection π : M → N {displaystyle pi :M ightarrow N} to the quotient space N = M / G {displaystyle N=M/G} equipped with the quotient metric is a Riemannian submersion.For example, component-wise multiplication on S 3 ⊂ C 2 {displaystyle S^{3}subset mathbb {C} ^{2}} by the group of unit complex numbers yields the Hopf fibration. The sectional curvature of the target space of a Riemannian submersion can be calculated from the curvature of the total space by O'Neill's formula: where X , Y {displaystyle X,Y} are orthonormal vector fields on N {displaystyle N} , X ~ , Y ~ {displaystyle { ilde {X}},{ ilde {Y}}} their horizontal lifts to M {displaystyle M} , [ ∗ , ∗ ] {displaystyle } is the Lie bracket of vector fields and Z V {displaystyle Z^{V}} is the projection of the vector field Z {displaystyle Z} to the vertical distribution. In particular the lower bound for the sectional curvature of N {displaystyle N} is at least as big as the lower bound for the sectional curvature of M {displaystyle M} .