Riemannian Geometry

🌀Riemannian Geometry Unit 12 – Riemannian Submersions & Submanifolds

Riemannian submersions and submanifolds are key concepts in differential geometry. They provide powerful tools for understanding the relationships between manifolds and their geometric properties, allowing us to study curvature, geodesics, and other important features. These concepts have wide-ranging applications in physics, computer graphics, and data analysis. From Hopf fibrations to minimal surfaces, they offer insights into the structure of space and the behavior of physical systems, bridging the gap between abstract mathematics and real-world phenomena.

Key Concepts and Definitions

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

Riemannian Submersions: Basics

  • Riemannian submersion π:(M,gM)(B,gB)\pi: (M,g_M) \to (B,g_B) maps MM onto BB such that π\pi_* isometric on horizontal subspaces
    • Fibers π1(b)\pi^{-1}(b) are equidistant submanifolds of MM for each bBb \in B
  • Horizontal lift of a vector field XX on BB unique horizontal vector field X~\tilde{X} on MM projecting to XX under π\pi_*
  • Riemannian submersions preserve lengths and angles of horizontal vectors
    • Geodesics on BB lift to horizontal geodesics on MM
  • O'Neill's tensors AA and TT describe the integrability of the horizontal and vertical distributions
    • AA measures the obstruction to the integrability of the horizontal distribution H\mathcal{H}
    • TT measures the obstruction to the integrability of the vertical distribution V\mathcal{V}
  • Riemannian submersions with totally geodesic fibers have vanishing O'Neill tensor TT
  • Examples of Riemannian submersions include Hopf fibrations S2n+1CPnS^{2n+1} \to \mathbb{CP}^n and S4n+3HPnS^{4n+3} \to \mathbb{HP}^n

Submanifolds in Riemannian Geometry

  • Submanifold NN embedded in a Riemannian manifold (M,g)(M,g) inherits a Riemannian metric from the ambient space
  • Tangent space TpNT_pN naturally identified with a subspace of TpMT_pM for each pNp \in N
  • Normal space TpNT_pN^\perp orthogonal complement of TpNT_pN in TpMT_pM with respect to the Riemannian metric gg
  • Second fundamental form II(X,Y)=(XY)II(X,Y) = (\nabla_X Y)^\perp measures the extrinsic curvature of NN in MM
    • \nabla Levi-Civita connection of the ambient manifold MM
    • ()(\cdot)^\perp normal component of a vector in TpMT_pM
  • Submanifold totally geodesic if every geodesic in NN also a geodesic in MM
    • Equivalent to vanishing second fundamental form II0II \equiv 0
  • Minimal submanifolds critical points of the volume functional with vanishing mean curvature vector HH
  • 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
    • RN(X,Y,Z,W)=RM(X,Y,Z,W)+II(X,W),II(Y,Z)II(X,Z),II(Y,W)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
    • (XII)(Y,Z)=(YII)(X,Z)(\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 MM and BB related by O'Neill's formula involving the tensor AA
    • Vertical sectional curvatures of MM and the fibers related by O'Neill's formula involving the tensor TT
  • 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 KM(X,Y)=KB(πX,πY)34AXY2K^M(X,Y) = K^B(\pi_*X, \pi_*Y) - \frac{3}{4}\|A_X Y\|^2
    • Vertical sectional curvature KM(V,W)=KF(V,W)+TVW2K^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)=κ1κ2K = \det(S) = \kappa_1 \kappa_2, where κ1,κ2\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
    • RicN(X,X)=RicM(X,X)+H,II(X,X)II(X,)2\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
    • S1S3S2S^1 \hookrightarrow S^3 \to S^2, S1S2n+1CPnS^1 \hookrightarrow S^{2n+1} \to \mathbb{CP}^n, S3S4n+3HPnS^3 \hookrightarrow S^{4n+3} \to \mathbb{HP}^n
  • Isoparametric hypersurfaces in spheres as examples of submanifolds with constant principal curvatures
    • Clifford tori Sp(pp+q)×Sq(qp+q)Sp+q+1S^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 R3\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


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© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.