Submanifolds are like mini-worlds within bigger spaces. They inherit their structure from the larger space but have their own unique features. This section dives into how these smaller worlds fit into the bigger picture.
We'll look at how submanifolds get their shape and measurements from the larger space. We'll also explore special tools, like the Gauss map, that help us understand how these mini-worlds curve and bend in the bigger space.
Submanifolds and Induced Geometry
Defining Submanifolds and Their Properties
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Submanifold refers to a subset of a larger manifold with its own manifold structure
Inherits smooth structure and topology from the ambient manifold
Can be defined locally as the zero set of smooth functions (implicit definition)
Alternatively described using a parametrization or embedding (explicit definition)
Dimension of a submanifold is always less than or equal to the dimension of the ambient manifold
Classified as regular or immersed based on the properties of the inclusion map
Induced Metric and First Fundamental Form
arises naturally on a submanifold from the metric of the ambient manifold
Represents the restriction of the ambient metric to the tangent spaces of the submanifold
Mathematically expressed as ginduced(X,Y)=gambient(dF(X),dF(Y)) where F is the inclusion map
encodes the induced metric in local coordinates
Denoted as I=gijdxidxj where gij are the components of the induced metric
Determines intrinsic geometry of the submanifold (lengths, angles, areas)
Isometric Embeddings and Their Significance
Isometric embedding preserves the metric structure when mapping one manifold into another
Satisfies the condition ⟨X,Y⟩M=⟨dF(X),dF(Y)⟩N for all tangent vectors X and Y
Crucial in studying the relationship between intrinsic and extrinsic geometry
Nash embedding theorem guarantees the existence of isometric embeddings for Riemannian manifolds into Euclidean spaces
Applications include visualizing abstract manifolds and analyzing their geometric properties
Normal Bundle and Gauss Map
Normal Bundle Structure and Properties
Normal bundle consists of all vectors orthogonal to the tangent spaces of a submanifold
Defined as NpM={v∈TpN:⟨v,w⟩=0 for all w∈TpM} at each point p
Complement to the tangent bundle in the ambient
Dimension equals the codimension of the submanifold in the ambient manifold
Smooth vector bundle structure inherited from the ambient manifold
Crucial in studying the extrinsic geometry of submanifolds
Gauss Map and Its Geometric Interpretation
Gauss map assigns to each point on a submanifold its unit normal vector
For hypersurfaces, maps into the unit sphere of the ambient space
Formally defined as G:M→Sn,p↦ν(p) where ν is a unit normal vector field
Measures how the submanifold curves in the ambient space
Differential of the Gauss map relates to the shape operator
Generalizes to higher codimension using the Grassmannian manifold
Unit Normal Vector and Its Role
Unit normal vector represents the direction perpendicular to the tangent space at each point
Normalized to have unit length with respect to the ambient metric
For orientable submanifolds, can be chosen consistently to define a global normal vector field
Plays a crucial role in defining the and shape operator
Used in the Gauss formula to decompose ambient covariant derivatives
Essential in formulating the Gauss-Bonnet theorem for surfaces
Shape Operator and Weingarten Equation
Shape Operator Definition and Properties
Shape operator quantifies how the unit normal vector changes along the submanifold
Defined as Sp(X)=−∇Xν where ∇ is the ambient connection and ν is the unit normal
Self-adjoint linear operator on the tangent space at each point
Eigenvalues of the shape operator are the principal curvatures
Trace of the shape operator gives the
Determinant of the shape operator relates to the Gaussian curvature for surfaces
Weingarten Equation and Its Implications
Weingarten equation relates the differential of the Gauss map to the shape operator
Expressed as dGp(X)=−Sp(X) for tangent vectors X
Establishes the fundamental link between the extrinsic and intrinsic geometry
Allows computation of the shape operator through the Gauss map
Crucial in deriving the Gauss-
Generalizes to higher codimension using the second fundamental form
Second Fundamental Form and Curvature Analysis
Second fundamental form measures the extrinsic curvature of a submanifold
Defined as II(X,Y)=⟨S(X),Y⟩=−⟨∇XY,ν⟩ for tangent vectors X and Y
Symmetric bilinear form on the tangent space at each point
Local expression in coordinates: II=hijdxidxj where hij are the coefficients
Relates to the shape operator via ⟨S(X),Y⟩=II(X,Y)
Used to classify points on a surface as elliptic, hyperbolic, or parabolic
Key Terms to Review (16)
Codazzi equations: Codazzi equations are a set of mathematical conditions that relate the second fundamental form of a submanifold to the geometry of the ambient space. These equations ensure the compatibility of the intrinsic and extrinsic geometries of submanifolds, highlighting how curvature behaves when moving along the surface. They play a crucial role in understanding how submanifolds sit inside their ambient spaces and help establish relationships between different types of curvature.
Embedded submanifold: An embedded submanifold is a subset of a Riemannian manifold that is itself a manifold, with a structure that allows it to fit nicely within the larger manifold. This means it retains its manifold properties while being equipped with an induced Riemannian metric, making it possible to study the geometry of both the submanifold and the ambient space. Understanding embedded submanifolds is crucial for analyzing the geometric relationships between different manifolds and for studying properties such as curvature and distance within the context of Riemannian geometry.
First Fundamental Form: The first fundamental form is a mathematical construct that encodes the intrinsic geometry of a surface embedded in a higher-dimensional space. It describes how distances and angles are measured on the surface, providing essential information about its shape and curvature through the metric tensor associated with the surface.
Gauss' Equations: Gauss' Equations are a set of formulas that relate the geometry of a surface to the curvature of that surface in Riemannian Geometry. These equations provide critical information about how a surface bends within the surrounding space and link the intrinsic geometry of a surface with its extrinsic properties, particularly in the context of submanifolds and induced geometry.
Geometric Measure Theory: Geometric measure theory is a branch of mathematics that combines concepts from geometry and measure theory to study geometric properties of sets and functions in a rigorous way. It provides tools for analyzing subsets of Euclidean spaces and manifolds, focusing on generalizing notions like area and volume to more complex structures such as fractals and non-smooth spaces. This theory is vital for understanding submanifolds and induced geometries as well as influencing recent developments in geometric analysis.
Hypersurface: A hypersurface is a high-dimensional generalization of a surface, defined as a subset of a manifold that has one lower dimension than the ambient space. In this context, hypersurfaces can be thought of as submanifolds that help in understanding the geometry and topology of the surrounding manifold. They are essential for studying induced metrics and curvature properties from the ambient space to the lower-dimensional structure.
Immersed submanifold: An immersed submanifold is a subset of a manifold that, while it may not be embedded, locally resembles the manifold around it in terms of its structure and differentiability. This concept is crucial because it allows for the study of lower-dimensional structures within higher-dimensional manifolds while still maintaining a degree of geometric and analytical properties relevant to the larger manifold.
Induced metric: An induced metric is a way to define a Riemannian metric on a submanifold by pulling back the metric from the ambient space where the submanifold is embedded. It captures the intrinsic geometric properties of the submanifold, allowing one to measure distances and angles solely based on the structure of the submanifold itself, rather than the larger space it resides in. This concept highlights the relationship between a manifold and its surrounding environment.
Local chart: A local chart is a mathematical tool used in differential geometry that provides a way to describe a subset of a manifold using coordinate systems. It consists of a homeomorphism from an open subset of the manifold to an open subset of Euclidean space, which allows for analysis and manipulation of geometric structures locally. Local charts are essential for understanding the properties of submanifolds and their induced geometries, as they facilitate the transition between the abstract manifold and more familiar Euclidean spaces.
Mean Curvature: Mean curvature is a measure of the curvature of a surface in a Riemannian manifold, specifically defined as the average of the principal curvatures at a given point. It helps characterize how a surface bends in space and plays a crucial role in the study of submanifolds, where it indicates the way an embedded surface interacts with its ambient space, revealing important geometric properties and the behavior of surfaces under variations.
Second Fundamental Form: The second fundamental form is a mathematical tool used to describe the intrinsic curvature of a submanifold embedded in a Riemannian manifold. It captures how the submanifold bends within the ambient space and provides crucial information about its geometry, such as how it deviates from being flat. Understanding this concept is key when looking at examples of Riemannian manifolds, exploring the geometry induced on submanifolds, and analyzing properties like mean curvature.
Sphere as a submanifold: A sphere as a submanifold refers to a mathematical structure that represents a set of points in a higher-dimensional space, constrained to maintain a constant distance from a central point. This concept illustrates how spheres can exist within larger Euclidean spaces and have their own intrinsic geometric properties, which can be examined through the lens of induced geometry.
Tangent Space: The tangent space at a point on a manifold is a vector space that consists of all possible tangent vectors at that point. It captures the local linear approximation of the manifold and connects to various concepts such as differentiability and geometry, allowing us to analyze how curves and surfaces behave in the vicinity of that point.
Tubular Neighborhood Theorem: The Tubular Neighborhood Theorem states that around every embedded submanifold of a Riemannian manifold, there exists a neighborhood that is diffeomorphic to a normal bundle. This theorem provides a way to visualize submanifolds within larger manifolds, allowing us to understand the geometry of the submanifold in relation to the ambient space. The theorem is fundamental for establishing properties such as the induced metric on the submanifold and understanding how local geometry behaves near the submanifold.
Variational methods: Variational methods are mathematical techniques used to find extrema of functionals, often involving calculus of variations. They are employed to analyze geometric and physical problems, allowing for the optimization of shapes and surfaces, and play a crucial role in understanding the properties of submanifolds and spectral geometry.
Whitney Embedding Theorem: The Whitney Embedding Theorem states that any smooth manifold can be embedded into a Euclidean space of sufficiently high dimension. This means that for every smooth manifold, there exists a smooth injection (embedding) into some $$ ext{R}^n$$, where $$n$$ is determined by the dimension of the manifold and the number of dimensions needed to avoid self-intersections.