Induced metrics on submanifolds are a key concept in Metric Differential Geometry. They allow us to study the intrinsic geometry of subsets within larger spaces, connecting local properties to global structures.
By examining how metrics from ambient spaces transfer to submanifolds, we gain insights into curvature, geodesics, and other geometric features. This topic bridges intrinsic and extrinsic perspectives, revealing the interplay between a 's internal structure and its .
Submanifolds in metric spaces
Submanifolds are subsets of a metric space that locally resemble Euclidean space and inherit the metric structure from the ambient space
The study of submanifolds is a central topic in Metric Differential Geometry as it allows for the analysis of geometric properties and structures within a larger space
Submanifolds provide a framework for understanding the interplay between the intrinsic geometry of the submanifold and the extrinsic geometry induced by the ambient space
Definition of submanifolds
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A subset M of a metric space X is called a submanifold if for every point p∈M, there exists a neighborhood U⊂X of p and a homeomorphism ϕ:U→V⊂Rn such that ϕ(U∩M)=V∩(Rk×{0}) for some k≤n
The dimension of the submanifold M is defined as the dimension k of the Euclidean space Rk in the local parametrization
Submanifolds can be characterized by their local charts and transition maps, which are compatible with the subspace topology and the metric structure
Topological properties of submanifolds
Submanifolds inherit the topological properties of the ambient space, such as separability, connectedness, and compactness
The subspace topology on a submanifold coincides with the topology induced by the metric structure
Submanifolds can be classified according to their topological properties, such as being closed, open, or properly embedded
Smooth structures on submanifolds
A submanifold M of a smooth manifold X is called a smooth submanifold if the inclusion map i:M→X is a smooth immersion
Smooth submanifolds inherit the smooth structure from the ambient manifold, allowing for the study of differential geometric properties
The of a smooth submanifold can be identified with a subspace of the tangent space of the ambient manifold at each point
Riemannian metrics on submanifolds
Riemannian metrics on submanifolds describe the intrinsic geometry of the submanifold and its relationship to the ambient space
The study of Riemannian metrics on submanifolds is crucial for understanding curvature, geodesics, and other geometric properties
Submanifolds with induced Riemannian metrics provide a rich setting for exploring the interplay between intrinsic and extrinsic geometry
Restriction of ambient metric
Given a Riemannian manifold (X,g) and a submanifold M⊂X, the g to the tangent spaces of M defines a on the submanifold
The restricted metric, denoted by g∣M, is compatible with the smooth structure on the submanifold and induces an inner product on each tangent space
The restriction of the ambient metric allows for the study of the intrinsic geometry of the submanifold using tools from Riemannian geometry
Intrinsic vs extrinsic geometry
Intrinsic geometry refers to the properties of a submanifold that depend only on the induced Riemannian metric and not on the embedding in the ambient space (e.g., Gaussian curvature of a surface)
Extrinsic geometry describes the properties of a submanifold that depend on its embedding in the ambient space and the relationship between the intrinsic and ambient metrics (e.g., of a hypersurface)
The interplay between intrinsic and extrinsic geometry is a central theme in the study of submanifolds and their geometric properties
Induced Riemannian metrics
An induced Riemannian metric on a submanifold M is obtained by restricting the ambient metric to the tangent spaces of M and extending it to a smooth tensor field
The is uniquely determined by the embedding of the submanifold and is compatible with the submanifold topology and smooth structure
Induced Riemannian metrics allow for the study of geometric properties, such as curvature and geodesics, using techniques from Riemannian geometry
Induced metrics and isometric embeddings
Induced metrics and isometric embeddings are fundamental concepts in the study of submanifolds and their geometric properties
The relationship between induced metrics and isometric embeddings provides a framework for understanding the intrinsic geometry of submanifolds and their relationship to the ambient space
Characterizing induced metrics and isometric embeddings is essential for classifying submanifolds and studying their geometric properties
Definition of induced metric
Given a smooth map f:M→N between Riemannian manifolds (M,g) and (N,h), the induced metric on M is defined as f∗h, where (f∗h)(X,Y)=h(df(X),df(Y)) for all vector fields X,Y on M
The induced metric is a symmetric, positive-definite (0, 2)-tensor field on M that is compatible with the smooth structure and topology of the submanifold
The induced metric captures the intrinsic geometry of the submanifold as determined by the embedding map f
Isometric embeddings and immersions
An isometric embedding is a smooth map f:(M,g)→(N,h) between Riemannian manifolds such that f∗h=g, i.e., the induced metric on M coincides with the given metric g
An isometric immersion is a smooth map f:(M,g)→(N,h) that is an immersion (i.e., df is injective at each point) and satisfies f∗h=g
Isometric embeddings and immersions preserve the intrinsic geometry of the submanifold and provide a way to realize abstract Riemannian manifolds as submanifolds of other spaces
Characterization of induced metrics
A Riemannian metric g on a manifold M is said to be induced by an embedding if there exists a Riemannian manifold (N,h) and an embedding f:M→N such that f∗h=g
The Nash Embedding Theorem states that every Riemannian manifold can be isometrically embedded into a Euclidean space of sufficiently high dimension
Characterizing induced metrics helps in understanding the relationship between the intrinsic geometry of a submanifold and its extrinsic geometry in the ambient space
Geometric properties of submanifolds
The geometric properties of submanifolds, such as curvature and minimality, provide insights into the intrinsic and extrinsic geometry of the submanifold
Studying the geometric properties of submanifolds is essential for understanding their shape, behavior, and relationship to the ambient space
Metric Differential Geometry provides tools and techniques for analyzing the geometric properties of submanifolds and their variations
First and second fundamental forms
The of a submanifold is the induced Riemannian metric, which captures the intrinsic geometry of the submanifold
The is a symmetric bilinear form that measures the extrinsic curvature of the submanifold and its deviation from being totally geodesic
The first and second fundamental forms together describe the local geometry of the submanifold and its relationship to the ambient space
Gaussian and mean curvature
Gaussian curvature is an intrinsic measure of curvature for surfaces, defined as the product of the principal curvatures at each point
Mean curvature is an extrinsic measure of curvature for hypersurfaces, defined as the average of the principal curvatures at each point
Gaussian and mean curvature provide important information about the shape and behavior of submanifolds, such as the presence of umbilical points or the minimality of the submanifold
Minimal submanifolds and variations
A submanifold is called minimal if its mean curvature vanishes identically, i.e., it locally minimizes area among nearby submanifolds
are characterized by the vanishing of the first variation of the area functional, which leads to the equation
The study of minimal submanifolds and their variations is a rich area of research in Metric Differential Geometry, with applications in geometry, topology, and mathematical physics
Examples and applications
Concrete examples and applications of submanifolds help to illustrate the abstract concepts and techniques developed in Metric Differential Geometry
Studying specific classes of submanifolds, such as or , provides insights into the general theory and its implications
Applications of submanifold theory can be found in various areas of mathematics, physics, and engineering, such as the study of geometric flows, relativity theory, and computer graphics
Surfaces in Euclidean space
Surfaces in Euclidean space, such as spheres, cylinders, and tori, serve as basic examples of submanifolds and their geometric properties
The induced Riemannian metric on a surface in Euclidean space is related to the first fundamental form, which can be expressed in terms of the local parametrization
The Gaussian curvature of a surface in Euclidean space can be computed using the first and second fundamental forms, and it provides information about the intrinsic geometry of the surface
Hypersurfaces in Riemannian manifolds
Hypersurfaces are codimension-one submanifolds of Riemannian manifolds, such as hyperspheres in Euclidean space or hypersurfaces in Lorentzian manifolds
The geometry of hypersurfaces is governed by the induced Riemannian metric and the second fundamental form, which captures the extrinsic curvature
Hypersurfaces play a crucial role in the study of geometric flows, such as mean curvature flow, and in the formulation of the positive mass theorem in general relativity
Geodesic submanifolds and totally geodesic submanifolds
A submanifold is called geodesic if every geodesic in the submanifold with respect to the induced metric is also a geodesic in the ambient space
A submanifold is called totally geodesic if every geodesic in the ambient space that is tangent to the submanifold at one point remains in the submanifold
Geodesic and are important examples of submanifolds with special geometric properties, and they arise naturally in the study of symmetric spaces and homogeneous manifolds
Connections and parallel transport
Connections and are fundamental tools in Metric Differential Geometry for studying the intrinsic geometry of submanifolds and their relationship to the ambient space
The Levi-Civita connection on a submanifold is induced by the Levi-Civita connection on the ambient space and is compatible with the induced Riemannian metric
Parallel transport and geodesics on submanifolds are closely related to the intrinsic and extrinsic geometry of the submanifold, as described by the Gauss and Codazzi-Mainardi equations
Levi-Civita connection on submanifolds
The Levi-Civita connection on a Riemannian manifold is a unique torsion-free metric connection that is compatible with the Riemannian metric
The Levi-Civita connection on a submanifold is induced by the Levi-Civita connection on the ambient space and satisfies the properties of metric compatibility and torsion-freeness
The induced Levi-Civita connection on a submanifold allows for the study of parallel transport, geodesics, and curvature on the submanifold using techniques from Riemannian geometry
Parallel transport and geodesics
Parallel transport is a way of moving tangent vectors along curves in a manifold while preserving their angle and length with respect to the Riemannian metric
Geodesics are curves that minimize the distance between points and are characterized by the property that their tangent vectors remain parallel along the curve with respect to the Levi-Civita connection
The relationship between parallel transport and geodesics on submanifolds is governed by the Gauss and Codazzi-Mainardi equations, which relate the intrinsic and extrinsic curvature of the submanifold
Gauss and Codazzi-Mainardi equations
The relates the intrinsic curvature of a submanifold (as measured by the Riemann curvature tensor of the induced metric) to the extrinsic curvature (as measured by the second fundamental form) and the curvature of the ambient space
The is a compatibility condition between the first and second fundamental forms, expressing the symmetry of the second fundamental form with respect to the induced Levi-Civita connection
The Gauss and Codazzi-Mainardi equations provide a fundamental link between the intrinsic and extrinsic geometry of submanifolds and are essential tools in the study of submanifolds and their geometric properties
Submanifolds of special Riemannian manifolds
The study of submanifolds of special Riemannian manifolds, such as spaces of constant curvature, symmetric spaces, and Kähler manifolds, reveals important connections between the intrinsic geometry of the submanifold and the geometric structure of the ambient space
Submanifolds of special Riemannian manifolds often inherit additional geometric properties and symmetries from the ambient space, leading to interesting classes of submanifolds with rich geometric structures
Understanding the behavior of submanifolds in special Riemannian manifolds is important for applications in various areas of mathematics and physics, such as the study of minimal surfaces, Einstein manifolds, and harmonic maps
Submanifolds of spaces of constant curvature
Spaces of constant curvature, such as Euclidean spaces, spheres, and hyperbolic spaces, are Riemannian manifolds with constant sectional curvature
inherit special geometric properties, such as the existence of a parallel second fundamental form or the property of being totally umbilical
The study of submanifolds in spaces of constant curvature is closely related to the theory of isometric immersions and the classification of space forms
Submanifolds of symmetric spaces
Symmetric spaces are Riemannian manifolds with a rich algebraic and geometric structure, characterized by the existence of symmetries at each point that preserve the Riemannian metric
, such as Lie groups or Grassmannians, often inherit the symmetries of the ambient space and can be studied using techniques from representation theory and Lie theory
The geometry of submanifolds in symmetric spaces is related to the theory of Jordan algebras, isoparametric submanifolds, and the study of homogeneous spaces
Kähler submanifolds of Kähler manifolds
Kähler manifolds are complex manifolds equipped with a compatible Riemannian metric and a complex structure that satisfies the Kähler condition
Kähler submanifolds are complex submanifolds of Kähler manifolds that inherit the Kähler structure from the ambient space, i.e., the induced metric and complex structure satisfy the Kähler condition
The study of Kähler submanifolds is important in complex differential geometry and has applications in the theory of minimal surfaces, Einstein manifolds, and the geometry of moduli spaces
Key Terms to Review (31)
Area Element: The area element is a mathematical concept used to measure the infinitesimal area in the context of a manifold. It plays a crucial role in defining integration over surfaces and submanifolds, allowing us to compute quantities like area and volume within a given geometric setting.
Codazzi-Mainardi Equation: The Codazzi-Mainardi equation is a set of equations in differential geometry that relates the second fundamental form of a submanifold to its curvature and the ambient space's curvature. These equations are crucial in understanding how submanifolds curve within higher-dimensional spaces, providing insights into the intrinsic and extrinsic geometry of the submanifold. They play a significant role in describing the compatibility conditions for the embedding of submanifolds.
Diffeomorphism: A diffeomorphism is a smooth, invertible function between two differentiable manifolds that has a smooth inverse. This concept is crucial in understanding how manifolds relate to each other and allows for the comparison of their geometric structures. Diffeomorphisms preserve the manifold's differentiable structure, making them essential for analyzing properties like tangent spaces and induced metrics when considering submanifolds.
Embedding: In differential geometry, an embedding refers to a smooth and injective map from one manifold to another, allowing the first manifold to be treated as a submanifold within the second. This concept is essential when discussing submanifolds since it captures how lower-dimensional spaces can exist and interact within higher-dimensional ones, preserving their geometric structure. When a manifold is embedded, it inherits a metric from the ambient space, which allows us to study properties like lengths, angles, and curvature in the context of its surrounding environment.
First Fundamental Form: The first fundamental form is a mathematical concept that provides a way to measure lengths and angles on a surface. It is defined by the inner product of tangent vectors at each point on the surface, allowing us to derive important geometric properties such as distances and areas. This form connects deeply with concepts like curvature, induced metrics on submanifolds, and the geometric behavior of parametrized surfaces.
Gauss Equation: The Gauss Equation relates the intrinsic curvature of a surface to the extrinsic curvature as embedded in a higher-dimensional space. It plays a crucial role in understanding the geometry of submanifolds and provides essential information about how the curvature behaves when viewed from different perspectives.
Gauss's Theorem: Gauss's Theorem, also known as the Divergence Theorem, relates the flow of a vector field through a closed surface to the behavior of the vector field inside the surface. It establishes a profound connection between differential geometry and physics, particularly in the context of curvature and induced metrics. Understanding this theorem is crucial for analyzing how geometric properties of surfaces relate to their embedding in higher-dimensional spaces.
Geodesic Submanifolds: Geodesic submanifolds are subsets of a manifold where the geometry resembles that of the larger manifold but is constrained to the submanifold itself. Within these submanifolds, geodesics can be defined as curves that locally minimize distances, maintaining the same notions of curvature and metrics as the surrounding space. This concept plays a significant role in understanding how intrinsic geometries behave within a broader context, particularly when considering induced metrics on these submanifolds.
Geodesics on a submanifold: Geodesics on a submanifold are curves that locally minimize distance within the submanifold, acting as the generalization of straight lines in curved spaces. They are critical for understanding how distances and angles behave in the context of submanifolds, which are themselves defined by the induced metric from the ambient manifold. By analyzing geodesics, we can explore intrinsic properties of the submanifold while also appreciating their relationship with the larger space they inhabit.
Hypersurfaces in Riemannian Manifolds: Hypersurfaces in Riemannian manifolds are submanifolds of dimension one less than that of the ambient manifold, providing a way to study the geometry and topology of the surrounding space. These hypersurfaces allow us to explore various geometric properties, such as curvature, by inducing metrics from the larger manifold. By understanding how Riemannian structures behave on these lower-dimensional surfaces, we gain insights into the global properties of the manifold itself.
Induced Metric: An induced metric is a way to define the geometric properties of a submanifold by pulling back the metric from a larger manifold. It captures how distances and angles are measured within the submanifold by restricting the original metric's properties to it. This concept is crucial for understanding how curved spaces relate to their embedded counterparts, allowing for the analysis of geometrical structures that reside within higher-dimensional spaces.
Isometry: An isometry is a distance-preserving map between metric spaces, meaning it maintains the same distances between points before and after the mapping. This concept is crucial in understanding how different geometries relate to one another, particularly in how metrics can be induced on submanifolds, warped product metrics, and symmetric spaces, all while maintaining the structure of the original manifold.
Kähler submanifolds of Kähler manifolds: Kähler submanifolds are special types of submanifolds that exist within Kähler manifolds, which are complex manifolds equipped with a Kähler metric. These submanifolds inherit the Kähler structure from the ambient Kähler manifold, meaning they have both a compatible symplectic form and a Riemannian metric. This connection allows for rich geometric properties and facilitates the study of complex geometry through the lens of differential geometry.
Length of curves: The length of curves is a measure of the distance along a curve, calculated by integrating the speed of a parameterized curve over its domain. This concept is crucial when discussing induced metrics on submanifolds and the Riemannian distance function, as it provides a way to understand how lengths are defined and computed in various geometric contexts. By understanding the length of curves, one can grasp how distances are represented in curved spaces, allowing for deeper insights into the geometry of these manifolds.
Levi-Civita Connection on Submanifolds: The Levi-Civita connection on submanifolds is a unique connection that is compatible with the induced metric and is torsion-free. This means it preserves the geometric structure of the submanifold while allowing for the definition of parallel transport and geodesics within that submanifold. Its properties are essential for understanding the curvature and overall geometry of submanifolds in the context of differential geometry.
Mean Curvature: Mean curvature is a measure of the curvature of a surface at a point, defined as the average of the principal curvatures. It plays a crucial role in understanding the geometric properties of surfaces, including their shapes and stability, and is closely related to concepts like the first and second fundamental forms, Gaussian curvature, and minimal surfaces.
Minimal Submanifolds: Minimal submanifolds are those submanifolds that locally minimize volume or area in a given ambient manifold, characterized by having zero mean curvature. This concept is closely tied to induced metrics, as the properties of minimal submanifolds are deeply influenced by the geometry of the ambient space and the induced metric on the submanifold itself, impacting how distances and angles are measured.
Minimal Surface: A minimal surface is a surface that locally minimizes area for a given boundary, characterized by having zero mean curvature at every point. These surfaces arise naturally in various contexts, particularly in the study of geometric properties of manifolds and variational problems, linking them closely to fundamental forms, induced metrics, and curvature concepts in differential geometry.
Normal Bundle: The normal bundle of a submanifold is a vector bundle that captures the directions in which the submanifold can be displaced within its ambient manifold. It consists of all the vectors that are orthogonal to the tangent space of the submanifold at each point, providing essential information about the geometry of both the submanifold and the surrounding manifold. This concept is crucial for understanding how submanifolds interact with their environment, including properties like curvature and metrics.
Parallel Transport: Parallel transport is a method of moving vectors along a curve on a manifold while keeping them 'parallel' according to the manifold's connection. This process is crucial for understanding how vectors behave in curved spaces, and it ties into various concepts like induced metrics on submanifolds, covariant derivatives, and geodesics, helping to maintain the geometric structure of the manifold.
Pullback: In differential geometry, a pullback refers to the operation that takes a differential form defined on a manifold and allows it to be transported back to another manifold via a smooth map. This concept is crucial for understanding how properties of one manifold can relate to another, particularly when dealing with smooth manifolds, induced metrics on submanifolds, and conformal metrics. The pullback essentially enables us to analyze forms and functions in the context of their original manifolds, facilitating the study of geometric structures.
Restriction of the metric: The restriction of the metric refers to the process of taking the original metric defined on a manifold and limiting its application to a submanifold. This process allows one to analyze the geometry of the submanifold using the properties inherited from the larger manifold, providing insights into distances and angles within that smaller context.
Riemannian metric: A Riemannian metric is a smoothly varying positive definite inner product on the tangent spaces of a differentiable manifold, allowing one to measure distances and angles on that manifold. This concept forms the backbone of Riemannian geometry, enabling the exploration of curves, surfaces, and their properties in various contexts, from smooth manifolds to curvature concepts and beyond.
Second Fundamental Form: The second fundamental form is a mathematical object that describes the intrinsic curvature of a surface embedded in a higher-dimensional space. It provides crucial information about how the surface bends and curves, allowing us to analyze geometric properties such as curvature and the relationship between the surface and the ambient space.
Submanifold: A submanifold is a subset of a manifold that itself has the structure of a manifold, allowing it to be smoothly embedded within the larger manifold. This concept is crucial in understanding the local and global properties of manifolds, especially when exploring tangent spaces, induced metrics, and how lengths and volumes are defined on these lower-dimensional surfaces.
Submanifolds of Spaces of Constant Curvature: Submanifolds of spaces of constant curvature are subsets of manifolds that have a constant curvature, such as spheres or hyperbolic spaces, and retain their geometric properties within these larger manifolds. These submanifolds can inherit induced metrics, which influence how distances and angles are measured on them, leading to unique geometrical characteristics.
Submanifolds of Symmetric Spaces: Submanifolds of symmetric spaces are special types of submanifolds that inherit their geometric properties from a larger symmetric space, allowing them to maintain certain symmetries and structures. These submanifolds are crucial for understanding how local and global geometric properties interact, particularly through the concept of induced metrics, which help in measuring distances and angles within the submanifold based on the ambient space.
Surfaces in Euclidean Space: Surfaces in Euclidean space refer to two-dimensional manifolds that exist within a three-dimensional Euclidean setting. These surfaces can be visualized as the shape of an object, like a sphere or a plane, and play a crucial role in understanding geometry and the behavior of curves. They allow for the exploration of properties like curvature, area, and distances, which are important when considering how these surfaces interact with their surrounding space.
Tangent Space: The tangent space at a point on a manifold is a vector space that consists of all possible directions in which one can tangentially pass through that point. This concept allows us to generalize the notion of derivatives from calculus to the context of manifolds, enabling the study of how functions behave locally around points on these complex structures.
Totally Geodesic Submanifolds: Totally geodesic submanifolds are special types of submanifolds where any geodesic that starts in the submanifold remains within the submanifold for all time. This means that the second fundamental form of the submanifold vanishes, indicating that the submanifold is as 'flat' as it can be relative to the ambient manifold. They play a crucial role in understanding the geometry of submanifolds and how they interact with their surrounding space, particularly in the context of induced metrics.
Whitney Embedding Theorem: The Whitney Embedding Theorem states that every smooth manifold can be embedded into Euclidean space of sufficiently high dimension. This theorem is significant because it provides a way to visualize and analyze manifolds using familiar geometric concepts, allowing for the study of their properties in a more tangible context.