Riemannian Geometry Unit 12 Review: Riemannian Submersions & Submanifolds

Key Concepts and Definitions

• Riemannian manifold $M$ equipped with a Riemannian metric $g$ that defines an inner product on each tangent space $T_pM$
• Smooth map $\pi: M \to B$ between Riemannian manifolds $(M,g_M)$ and $(B,g_B)$
• Horizontal and vertical distributions $\mathcal{H}$ and $\mathcal{V}$ decompose the tangent space at each point of $M$
  • Horizontal distribution $\mathcal{H}$ consists of tangent vectors orthogonal to the fibers of $\pi$
  • Vertical distribution $\mathcal{V}$ tangent to the fibers of $\pi$
• Isometric submersion preserves the length of horizontal vectors under the projection $\pi_*: T_pM \to T_{\pi(p)}B$
• Ehresmann connection $\mathcal{H}$ provides a smooth choice of horizontal subspaces complementary to the vertical subspaces
• Second fundamental form $II$ measures the failure of a submanifold to be totally geodesic
• Mean curvature vector $H$ average of the second fundamental form over all tangent directions

Riemannian Submersions: Basics

• Riemannian submersion $\pi: (M,g_M) \to (B,g_B)$ maps $M$ onto $B$ such that $\pi_*$ isometric on horizontal subspaces
  • Fibers $\pi^{-1}(b)$ are equidistant submanifolds of $M$ for each $b \in B$
• Horizontal lift of a vector field $X$ on $B$ unique horizontal vector field $\tilde{X}$ on $M$ projecting to $X$ under $\pi_*$
• Riemannian submersions preserve lengths and angles of horizontal vectors
  • Geodesics on $B$ lift to horizontal geodesics on $M$
• O'Neill's tensors $A$ and $T$ describe the integrability of the horizontal and vertical distributions
  • $A$ measures the obstruction to the integrability of the horizontal distribution $\mathcal{H}$
  • $T$ measures the obstruction to the integrability of the vertical distribution $\mathcal{V}$
• Riemannian submersions with totally geodesic fibers have vanishing O'Neill tensor $T$
• Examples of Riemannian submersions include Hopf fibrations $S^{2n+1} \to \mathbb{CP}^n$ and $S^{4n+3} \to \mathbb{HP}^n$

Submanifolds in Riemannian Geometry

• Submanifold $N$ embedded in a Riemannian manifold $(M,g)$ inherits a Riemannian metric from the ambient space
• Tangent space $T_pN$ naturally identified with a subspace of $T_pM$ for each $p \in N$
• Normal space $T_pN^\perp$ orthogonal complement of $T_pN$ in $T_pM$ with respect to the Riemannian metric $g$
• Second fundamental form $II(X,Y) = (\nabla_X Y)^\perp$ measures the extrinsic curvature of $N$ in $M$
  • $\nabla$ Levi-Civita connection of the ambient manifold $M$
  • $(\cdot)^\perp$ normal component of a vector in $T_pM$
• Submanifold totally geodesic if every geodesic in $N$ also a geodesic in $M$
  • Equivalent to vanishing second fundamental form $II \equiv 0$
• Minimal submanifolds critical points of the volume functional with vanishing mean curvature vector $H$
• Examples of submanifolds include hypersurfaces, curves, and surfaces embedded in Euclidean spaces or Riemannian manifolds

Fundamental Theorems and Properties

• Gauss equation relates the intrinsic curvature of a submanifold to its extrinsic curvature and the curvature of the ambient space
  • $R^N(X,Y,Z,W) = R^M(X,Y,Z,W) + \langle II(X,W), II(Y,Z)\rangle - \langle II(X,Z), II(Y,W)\rangle$
• Codazzi-Mainardi equation compatibility condition between the second fundamental form and the Levi-Civita connection
  • $(\nabla_X II)(Y,Z) = (\nabla_Y II)(X,Z)$
• Fundamental theorem of Riemannian submersions relates the curvature of the total space, base space, and fibers
  • Horizontal sectional curvatures of $M$ and $B$ related by O'Neill's formula involving the tensor $A$
  • Vertical sectional curvatures of $M$ and the fibers related by O'Neill's formula involving the tensor $T$
• Bonnet-Myers theorem bounds the diameter of a complete Riemannian manifold with positive Ricci curvature
  • Applicable to total spaces of Riemannian submersions and ambient spaces of submanifolds
• Synge's theorem characterizes the fundamental group of compact oriented even-dimensional Riemannian manifolds with positive sectional curvature
  • Relevant for studying the topology of total spaces and ambient spaces

Curvature in Submersions and Submanifolds

• Sectional curvature of a Riemannian submersion related to the curvatures of the base and fibers via O'Neill's formulas
  • Horizontal sectional curvature $K^M(X,Y) = K^B(\pi_*X, \pi_*Y) - \frac{3}{4}\|A_X Y\|^2$
  • Vertical sectional curvature $K^M(V,W) = K^F(V,W) + \|T_V W\|^2$
• Positive (negative) curvature of the base and fibers implies positive (negative) curvature of the total space under certain conditions
• Gaussian curvature of a surface in Euclidean space given by the determinant of the shape operator
  • $K = \det(S) = \kappa_1 \kappa_2$, where $\kappa_1, \kappa_2$ are the principal curvatures
• Ricci curvature of a submanifold related to the Ricci curvature of the ambient space and the mean curvature vector
  • $\operatorname{Ric}^N(X,X) = \operatorname{Ric}^M(X,X) + \langle H, II(X,X)\rangle - \|II(X,\cdot)\|^2$
• Scalar curvature of a submanifold can be expressed using the scalar curvature of the ambient space, the mean curvature, and the norm of the second fundamental form
• Curvature estimates and comparison theorems for Riemannian submersions and submanifolds
  • Bounds on the curvature of the total space or ambient space yield bounds on the curvature of the base, fibers, or submanifold

Applications and Examples

• Hopf fibrations as examples of Riemannian submersions with various geometric properties
  • $S^1 \hookrightarrow S^3 \to S^2$, $S^1 \hookrightarrow S^{2n+1} \to \mathbb{CP}^n$, $S^3 \hookrightarrow S^{4n+3} \to \mathbb{HP}^n$
• Isoparametric hypersurfaces in spheres as examples of submanifolds with constant principal curvatures
  • Clifford tori $S^p(\sqrt{\frac{p}{p+q}}) \times S^q(\sqrt{\frac{q}{p+q}}) \subset S^{p+q+1}$
• Geodesic spheres and tubes as examples of umbilic hypersurfaces with constant mean curvature
• Minimal surfaces in Euclidean space $\mathbb{R}^3$ and other Riemannian manifolds
  • Catenoid, helicoid, Costa-Hoffman-Meeks surfaces
• Riemannian submersions in physics, such as Kaluza-Klein theory and Yang-Mills theory
  • Extra dimensions compactified as fibers of a submersion, yielding a lower-dimensional effective theory
• Submanifold theory in computer graphics and data analysis
  • Manifold learning, dimensionality reduction, and surface reconstruction

Connections to Other Areas of Geometry

• Relationship between Riemannian submersions and fiber bundles in differential topology
  • Ehresmann connection provides a horizontal distribution complementary to the vertical distribution
• Riemannian submersions as a tool for constructing new Riemannian manifolds with prescribed curvature properties
  • Cheeger deformations, warped products, and doubly-warped products
• Submanifold theory as a bridge between extrinsic and intrinsic geometry
  • Gauss-Bonnet theorem relates the Euler characteristic to the integral of Gaussian curvature
• Connections to complex and Kähler geometry through the study of complex submanifolds and holomorphic submersions
• Relationship to harmonic maps and minimal immersions in geometric analysis
• Applications of submersions and submanifolds in mathematical physics, such as general relativity and string theory
  • Spacetime as a Lorentzian manifold, with matter and energy modeled as submanifolds or fields

Advanced Topics and Current Research

• Foliations and Riemannian foliations as generalizations of Riemannian submersions
  • Singular Riemannian foliations, orbit-like foliations, and polar foliations
• Infinite-dimensional submersions and submanifolds in the context of Hilbert and Banach manifolds
  • Morse theory and the calculus of variations on infinite-dimensional manifolds
• Harmonic morphisms as a special class of Riemannian submersions preserving Laplace's equation
  • Characterizations, existence results, and applications in physics and analysis
• Calibrated geometries and special Lagrangian submanifolds in Calabi-Yau manifolds
  • Connections to mirror symmetry and string theory
• Mean curvature flow and other geometric flows for submanifolds
  • Singularity formation, classification, and applications in topology and physics
• Rigidity and stability results for Riemannian submersions and submanifolds under curvature assumptions
• Higher-order curvature invariants and their role in the study of submersions and submanifolds
• Applications of submersion and submanifold theory in data science, machine learning, and computer vision