📐Metric Differential Geometry Unit 8 – Submanifolds and Hypersurfaces in Geometry

Key Concepts and Definitions

• Manifold: A topological space that locally resembles Euclidean space near each point (e.g., a sphere or a torus)
• Submanifold: A subset of a manifold that is itself a manifold with the subspace topology
  • Inherits the differential structure from the ambient manifold
• Hypersurface: A submanifold of codimension 1, meaning its dimension is one less than the ambient manifold
• Tangent space: A vector space attached to each point of a manifold, consisting of all tangent vectors at that point
  • Tangent vectors represent instantaneous velocities of curves passing through the point
• Normal vector: A vector perpendicular to the tangent space at a given point on a submanifold
• Riemannian metric: A smooth, positive-definite, symmetric bilinear form on the tangent space at each point of a manifold
  • Allows for the measurement of lengths, angles, and curvature on the manifold

Submanifolds: Types and Properties

• Embedded submanifold: A submanifold that is a subset of the ambient manifold and inherits its topology and differential structure
  • The inclusion map is a smooth embedding
• Immersed submanifold: A manifold that is mapped into the ambient manifold by a smooth immersion, which may not be one-to-one
  • The immersion induces a topology and differential structure on the submanifold
• Closed submanifold: A submanifold that is a closed subset of the ambient manifold
• Properly embedded submanifold: An embedded submanifold that is also a closed subset of the ambient manifold
• Codimension: The difference between the dimension of the ambient manifold and the dimension of the submanifold
• Induced Riemannian metric: The restriction of the ambient manifold's Riemannian metric to the tangent spaces of the submanifold

Hypersurfaces: Special Case of Submanifolds

• Codimension 1: Hypersurfaces have a codimension of 1, meaning their dimension is one less than the ambient manifold
  • In an n-dimensional manifold, hypersurfaces are (n-1)-dimensional
• Orientability: Hypersurfaces can be orientable or non-orientable
  • Orientable hypersurfaces have a consistent choice of normal vector at each point (e.g., a sphere)
  • Non-orientable hypersurfaces do not have a consistent choice of normal vector (e.g., a Möbius strip)
• Level sets: Hypersurfaces can often be described as the level sets of smooth functions
  • For a smooth function $f: M \to \mathbb{R}$, the level set $f^{-1}(c)$ for a regular value $c$ is a hypersurface
• Mean curvature: The trace of the second fundamental form, measuring the average curvature of a hypersurface at a point
• Gaussian curvature: The determinant of the second fundamental form, measuring the intrinsic curvature of a hypersurface at a point

Tangent Spaces and Normal Vectors

• Tangent space: A vector space attached to each point of a submanifold, consisting of all tangent vectors to the submanifold at that point
  • Tangent vectors can be represented as derivatives of curves lying on the submanifold
• Basis for tangent space: A set of linearly independent tangent vectors that span the tangent space at a point
  • For an n-dimensional submanifold, the tangent space has dimension n
• Normal space: The orthogonal complement of the tangent space at a point on a submanifold
  • Consists of all vectors perpendicular to the tangent space
• Unit normal vector: A normal vector of length 1, used to define the orientation of a hypersurface
• Orthogonal projection: The projection of a vector onto the tangent space or normal space of a submanifold
  • Allows for the decomposition of vectors into tangential and normal components

Curvature of Submanifolds

• First fundamental form: The induced Riemannian metric on a submanifold, measuring intrinsic properties like length and angle
  • Obtained by restricting the ambient manifold's Riemannian metric to the tangent spaces of the submanifold
• Second fundamental form: A quadratic form that measures the extrinsic curvature of a submanifold
  • Relates the change in the normal vector to the tangent vectors of the submanifold
• Principal curvatures: The eigenvalues of the second fundamental form, representing the maximum and minimum curvatures at a point
• Gaussian curvature: The product of the principal curvatures, measuring the intrinsic curvature of a submanifold at a point
  • Can be computed using only the first fundamental form (Theorema Egregium)
• Mean curvature: The average of the principal curvatures, measuring the extrinsic curvature of a submanifold at a point
  • Related to the trace of the second fundamental form
• Geodesic curvature: The curvature of a curve lying on a submanifold, measuring its deviation from a geodesic

Embedding and Immersion Theorems

• Whitney embedding theorem: Every smooth n-dimensional manifold can be smoothly embedded in $\mathbb{R}^{2n}$
  • Provides a upper bound on the codimension required for embedding
• Nash embedding theorem: Every Riemannian manifold can be isometrically embedded in a Euclidean space of sufficiently high dimension
  • The embedding preserves the Riemannian metric and all intrinsic properties of the manifold
• Immersion theorem: Every smooth n-dimensional manifold can be smoothly immersed in $\mathbb{R}^{2n-1}$
  • Immersions may have self-intersections, unlike embeddings
• Compact submanifolds: Embedding and immersion results often require the submanifold to be compact (closed and bounded)
• Codimension bounds: The theorems provide upper bounds on the codimension required for embeddings or immersions
  • Lower codimensions may be sufficient for specific manifolds or submanifolds

Applications in Physics and Engineering

• General relativity: Spacetime is modeled as a 4-dimensional Lorentzian manifold, with gravity described by its curvature
  • Geodesics on the spacetime manifold represent the paths of free-falling particles
• String theory: Submanifolds called branes play a crucial role in string theory, representing the higher-dimensional objects on which strings can end
• Mechanics: Configuration spaces of mechanical systems often have the structure of a manifold or submanifold
  • Constraints in the system can be modeled as submanifolds of the configuration space
• Computer graphics: Surfaces in 3D graphics are often represented as 2-dimensional submanifolds (e.g., triangle meshes)
  • Curvature information is used for shading, texture mapping, and other visual effects
• Robotics: The configuration space of a robot arm is a manifold, with each point representing a possible position and orientation of the arm
  • Path planning involves finding geodesics or other optimal paths on this manifold

Advanced Topics and Current Research

• Mean curvature flow: A geometric evolution equation that deforms a submanifold in the direction of its mean curvature vector
  • Used in image processing, surface fairing, and the study of minimal surfaces
• Ricci flow: A geometric evolution equation that deforms a Riemannian metric on a manifold based on its Ricci curvature
  • Played a key role in the proof of the Poincaré conjecture by Perelman
• Calibrated geometries: Special classes of submanifolds that minimize volume within their homology class
  • Examples include complex submanifolds, special Lagrangian submanifolds, and associative submanifolds
• Symplectic geometry: The study of symplectic manifolds, which have a closed, nondegenerate 2-form called the symplectic form
  • Lagrangian submanifolds are a key object of study in symplectic geometry
• Gauge theory: The study of connections on principal bundles and their curvature
  • Yang-Mills equations and instantons are important examples of gauge-theoretic equations on submanifolds