🌀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.
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)=κ1κ2, where κ1,κ2 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
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
S1↪S3→S2, S1↪S2n+1→CPn, S3↪S4n+3→HPn
Isoparametric hypersurfaces in spheres as examples of submanifolds with constant principal curvatures
Clifford tori Sp(p+qp)×Sq(p+qq)⊂Sp+q+1
Geodesic spheres and tubes as examples of umbilic hypersurfaces with constant mean curvature
Minimal surfaces in Euclidean space R3 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