Riemannian Submersions — Mathematics Knowledge System

Overview

A Riemannian submersion is a smooth surjective map π: (M, g_M) → (B, g_B) between Riemannian manifolds whose restriction to horizontal vectors (orthogonal complements to fibers) is a linear isometry at every point. Riemannian submersions generalize the projection from a principal bundle to the quotient, and include the Hopf fibration S³ → S². O'Neill's formula relates the curvature of M to that of B via the A-tensor, enabling curvature computations on orbit spaces and fiber bundles.

Intuition

In a Riemannian submersion, the total space M fibers over the base B, and the horizontal directions (orthogonal to fibers) are metrically equivalent to the base directions. The map does not distort horizontal distances: if you move horizontally in M, it corresponds isometrically to moving in B. The fibers can be curved, and the interaction between horizontal and vertical directions (measured by O'Neill's tensor A) contributes to the curvature of B.

Formal Definition

Definition

Let π: (M, g_M) → (B, g_B) be a smooth surjective map. For each p ∈ M, define the vertical subspace V_p = ker(dπ_p) and the horizontal subspace H_p = V_p^⊥ (orthogonal complement in T_pM). π is a Riemannian submersion if dπ_p|_{H_p}: H_p → T_{π(p)}B is a linear isometry for all p.

\mathcal{V}_p = \ker(d\pi_p), \quad \mathcal{H}_p = \mathcal{V}_p^\perp \subseteq T_pM

Vertical and horizontal subbundles

d\pi_p|_{\mathcal{H}_p}: \mathcal{H}_p \xrightarrow{\sim} T_{\pi(p)}B \quad \text{isometry}

Isometry condition on horizontal subspace

A_X Y = \mathcal{V}(\nabla_{\mathcal{H}X} \mathcal{H}Y) + \mathcal{H}(\nabla_{\mathcal{H}X} \mathcal{V}Y)

Measures the failure of horizontal distribution to be integrable

O'Neill A-tensor (integrability tensor)

Theorems

Theorem 1: O'Neill's Formula

\text{For a Riemannian submersion } \pi: M \to B \text{ and horizontal unit vectors } X, Y \in \mathcal{H}_p: \newline K^B(d\pi X, d\pi Y) = K^M(X,Y) + 3|A_X Y|^2.

Theorem 2: Geodesic Lifting

\text{A horizontal geodesic in } M \text{ (one whose velocity is always horizontal) projects to a geodesic in } B. \text{ Conversely, every geodesic in } B \text{ lifts to a horizontal geodesic in } M.

Worked Examples

1
S³ has constant sectional curvature K = 1. S²(1/2) (sphere of radius 1/2) has constant sectional curvature K = 4 (since K = 1/r² = 1/(1/2)² = 4).
$K(S^3) = 1, \quad K(S^2(1/2)) = 4$
2
O'Neill's formula: K^B = K^M + 3|A_XY|². So 4 = 1 + 3|A_XY|², giving |A_XY|² = 1.
$4 = 1 + 3|A_X Y|^2 \implies |A_X Y|^2 = 1$
3
The A-tensor is nonzero (A_XY ≠ 0 for orthonormal horizontal X,Y), meaning the horizontal distribution of the Hopf fibration is non-integrable (as expected from the non-trivial bundle topology).
$|A_X Y| = 1 \neq 0$
4
O'Neill's formula shows: K^B = K^M + (non-negative term). So the base always has higher (or equal) sectional curvature in horizontal directions than the total space. The non-integrability of horizontal directions increases curvature going from M to B.
$K^B \geq K^M \text{ (O'Neill inequality)}$

✓ Answer

The Hopf fibration S³ → S²(1/2) satisfies O'Neill: K^B = K^M + 3|AXY|² = 1 + 3 = 4. The base has higher curvature than the total space; the A-tensor (with |AXY| = 1) encodes the non-integrability of the horizontal distribution.

Practice Problems

Mediumfree response

State O'Neill's formula and explain the geometric meaning of the A-tensor.

Hardfree response

Give an example of a Riemannian submersion where the fibers are not totally geodesic, and explain what that means.

Quiz

Summary

A Riemannian submersion π: (M,g_M) → (B,g_B) has dπ|_H an isometry at each point, where H is the horizontal complement to the fibers.
Horizontal geodesics in M project to geodesics in B; every geodesic in B lifts horizontally.
O'Neill's formula: K^B = K^M + 3|AXY|² shows the base has higher curvature than the total space.
The A-tensor measures non-integrability of the horizontal distribution — it vanishes iff the fibration is locally trivial.
Key examples: Hopf fibrations S^(2n+1) → ℂPⁿ, SO(n+1) → Sⁿ, and more general principal bundle projections.