Riemannian submersions connect two Riemannian manifolds through a that preserves certain geometric properties. They allow us to study relationships between metrics and curvatures of different manifolds, providing insights into their structure.
These maps split the tangent space of the domain into vertical and horizontal spaces, with the acting as an isometry on the horizontal space. This framework enables us to analyze complex geometric relationships and construct new manifolds with specific properties.
Definition of Riemannian submersions
Riemannian submersions are a fundamental concept in Riemannian geometry that relate the geometry of two Riemannian manifolds via a smooth map
They provide a way to study the relationship between the metrics and curvatures of the domain and codomain manifolds
Smooth surjective maps
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A is a smooth map π:(M,gM)→(N,gN) between Riemannian manifolds that is surjective
The map π is required to be a submersion, meaning its differential dπ is surjective at every point
Surjectivity ensures that every point in the codomain N is the image of some point in the domain M
Riemannian metrics on domain and codomain
The domain manifold M is equipped with a gM, which defines an inner product on each tangent space
Similarly, the codomain manifold N has a Riemannian metric gN
The metrics allow for the measurement of lengths, angles, and volumes on the respective manifolds
Isometric on horizontal space
At each point p∈M, the tangent space TpM splits into the Vp and the horizontal space Hp
The Riemannian submersion π is required to be an isometry when restricted to the horizontal space
This means that for any horizontal vectors X,Y∈Hp, we have gM(X,Y)=gN(dπ(X),dπ(Y))
Horizontal and vertical distributions
The vertical distribution V consists of the kernel of dπ at each point, representing the tangent spaces to the fibers of π
The H is the orthogonal complement of V with respect to the metric gM
The distributions V and H are smooth and provide a decomposition of the tangent bundle TM
Properties of Riemannian submersions
Riemannian submersions exhibit several important properties that relate the geometry of the domain and codomain manifolds
These properties involve the rank and nullity of the submersion, curvature relations, and the geometry of the fibers
Rank and nullity
The rank of a Riemannian submersion π at a point p∈M is the dimension of the horizontal space Hp
The nullity of π at p is the dimension of the vertical space Vp, which equals the dimension of the π−1(π(p))
The rank and nullity satisfy the relation rank(π)+nullity(π)=dim(M)
O'Neill's tensors
, denoted by A and T, measure the integrability of the horizontal and vertical distributions
The tensor A is the obstruction to the integrability of the horizontal distribution H
The tensor T is the second fundamental form of the fibers and measures their extrinsic curvature
Fundamental equations
The relate the curvatures of the domain, codomain, and fibers of a Riemannian submersion
The O'Neill formulas express the sectional curvatures of M and N in terms of the A and T tensors
These equations provide insights into how the geometry of the domain and codomain are related
Geometry of fibers
The fibers of a Riemannian submersion π, given by π−1(y) for y∈N, are submanifolds of M
Each fiber inherits a Riemannian metric from the restriction of gM to the vertical space
The geometry of the fibers, such as their curvature and topology, plays a crucial role in understanding the overall structure of the submersion
Totally geodesic fibers
A fiber of a Riemannian submersion is called totally geodesic if every geodesic in the fiber is also a geodesic in the M
have vanishing second fundamental form, meaning T≡0
Riemannian submersions with totally geodesic fibers have particularly nice properties and are often studied in geometry and physics
Riemannian submersions vs Riemannian isometries
Riemannian submersions and Riemannian isometries are both important maps between Riemannian manifolds
While they share some similarities, there are key differences between the two concepts
Similarities in preserving metrics
Both Riemannian submersions and isometries preserve certain metric properties
Riemannian submersions are isometric on the horizontal space, meaning they preserve the length of horizontal curves
Riemannian isometries preserve the Riemannian metric on the entire tangent space, not just the horizontal space
Differences in surjectivity and injectivity
Riemannian submersions are surjective maps, ensuring that every point in the codomain is the image of some point in the domain
Riemannian isometries, on the other hand, are not necessarily surjective and can be injective (one-to-one)
Isometries preserve the Riemannian distance between points, while submersions may not preserve distances between points in different fibers
Constructions using Riemannian submersions
Riemannian submersions are a versatile tool for constructing new Riemannian manifolds with desired properties
They provide a way to build manifolds with prescribed geometry and symmetries
Warped product metrics
are a generalization of product metrics, where the metric on one factor is scaled by a function
Riemannian submersions can be used to construct warped product metrics, with the base manifold being the codomain and the fibers being the other factor
Warped products have applications in general relativity, such as in the study of spacetime geometries
Projective spaces
Projective spaces, such as real and complex projective spaces, can be constructed using Riemannian submersions
The , which is a Riemannian submersion from a sphere to a projective space, provides a geometric realization of these spaces
Projective spaces are important in algebraic geometry and have connections to gauge theory and physics
Hopf fibrations
The Hopf fibrations are a family of Riemannian submersions between spheres of different dimensions
The most well-known Hopf fibration is the submersion from the 3-sphere to the 2-sphere, with fibers being circles
Hopf fibrations have important applications in topology, geometry, and mathematical physics
Homogeneous spaces
are manifolds that admit a transitive action by a Lie group
Riemannian submersions can be used to construct homogeneous spaces as quotients of Lie groups
Many important manifolds in geometry and physics, such as Grassmannians and Stiefel manifolds, are homogeneous spaces
Applications of Riemannian submersions
Riemannian submersions have found numerous applications in various areas of mathematics and physics
They provide a powerful framework for studying the geometry and topology of manifolds and their relationships
Submanifold geometry
Riemannian submersions are closely related to the study of submanifolds in Riemannian geometry
The fibers of a Riemannian submersion are submanifolds of the domain manifold
Properties of the submersion, such as the O'Neill tensors and curvature relations, provide information about the geometry of the submanifolds
Quotient manifolds
Riemannian submersions can be used to construct by identifying points in the fibers
The metric on the quotient manifold is induced by the submersion, making it a Riemannian manifold
Quotient manifolds arise naturally in the study of symmetries and group actions on manifolds
Kaluza-Klein theory
is a geometric approach to unifying gravity and electromagnetism by considering higher-dimensional spacetimes
Riemannian submersions play a crucial role in Kaluza-Klein theory, with the extra dimensions being the fibers of the submersion
The geometry of the fibers and the base manifold determine the properties of the unified theory
Yang-Mills theory
is a gauge theory that describes the dynamics of non-Abelian gauge fields
Riemannian submersions provide a geometric framework for studying Yang-Mills theory
The principal bundles and connections in Yang-Mills theory can be described using Riemannian submersions and their associated structures
Examples of Riemannian submersions
There are many important and illustrative examples of Riemannian submersions in geometry and physics
These examples demonstrate the wide range of applications and the richness of the theory
Canonical projection of a product manifold
The canonical projection from a product manifold M×N onto one of its factors, such as π:M×N→M, is a Riemannian submersion
The fibers of this submersion are the copies of the other factor N
This example serves as a prototype for more general constructions of Riemannian submersions
Hopf map between spheres
The Hopf map is a Riemannian submersion from the unit sphere S2n+1 to the complex projective space CPn
The fibers of the Hopf map are great circles, which are totally geodesic submanifolds of the sphere
The Hopf map has important applications in topology, such as in the study of homotopy groups and characteristic classes
Twistor fibrations
Twistor theory, introduced by Roger Penrose, provides a geometric approach to studying solutions of certain differential equations
Twistor fibrations are Riemannian submersions that relate twistor spaces to base manifolds, such as the fibration from the twistor space of a 4-manifold to the manifold itself
Twistor fibrations have applications in mathematical physics, particularly in the study of Yang-Mills equations and integrable systems
Riemannian submersions in physics
Riemannian submersions have found numerous applications in theoretical physics
In general relativity, submersions are used to construct models of spacetime with extra dimensions, such as in Kaluza-Klein theory
In gauge theory, submersions provide a geometric framework for studying the geometry of gauge fields and their associated bundles
Riemannian submersions also appear in the study of supergravity, string theory, and other areas of mathematical physics
Key Terms to Review (26)
Base Space: The base space refers to the underlying manifold or geometric space from which more complex structures, such as fiber bundles or warped products, are constructed. It serves as the foundational layer upon which additional geometric or topological features are built, allowing for a structured approach to studying various mathematical properties and relationships in differential geometry.
Differentiable Structure: A differentiable structure on a manifold is a collection of charts that are compatible with each other, allowing for the smooth transition between local coordinate systems. This structure enables the application of calculus on manifolds, defining concepts such as smooth functions, tangent spaces, and differentiability. The consistency of these charts through transition maps is crucial for understanding how local properties relate to the manifold's global characteristics.
Ehresmann Connection: An Ehresmann connection is a mathematical structure that provides a way to compare tangent spaces of different fibers in a fiber bundle, facilitating the definition of parallel transport and connections between the base and total spaces. This concept plays a crucial role in understanding geometric structures like Riemannian submersions, where it helps preserve the geometry of the fibers while allowing for the projection onto the base manifold. Additionally, it offers a systematic method for discussing curvature and connections in differential geometry.
Fiber: In differential geometry, a fiber refers to the pre-image of a point in the base space of a fibration, representing a collection of points in the total space that map to that point. This concept is crucial for understanding how different geometric structures interact and can be visualized, particularly in warped product metrics and Riemannian submersions where fibers illustrate the relationship between the total space and base space.
Fiber Bundle: A fiber bundle is a structure that consists of a base space, a total space, and a fiber that varies smoothly over the base. This setup allows for a way to analyze complex spaces by breaking them down into simpler pieces, where each piece resembles the fiber itself. Fiber bundles play a crucial role in understanding concepts like tangent spaces, Riemannian submersions, homogeneous spaces, geometric mechanics, and parallel transport.
Fundamental Equations: Fundamental equations are essential mathematical expressions that encapsulate the core relationships between various geometric structures within a Riemannian manifold. They serve as the backbone for understanding the intrinsic and extrinsic properties of these manifolds, particularly in the context of Riemannian submersions, where they describe how geometric features behave under projection from one manifold to another.
Generalized submersion theorem: The generalized submersion theorem provides a framework for understanding how certain smooth maps between Riemannian manifolds behave in terms of their geometric properties. It essentially states that under specific conditions, the image of a Riemannian submersion retains certain Riemannian structures, leading to a well-defined geometry on the quotient manifold. This concept is vital for studying the relationship between the geometry of the total space and the geometry of the base space in Riemannian submersions.
Geometry of fibers: The geometry of fibers refers to the study of the geometric properties and structures of fibers in a fiber bundle, particularly in the context of Riemannian submersions. This concept emphasizes how the geometry of the total space influences the properties of the fibers, including curvature and distance, while maintaining a consistent relationship with the base space.
Homogeneous spaces: Homogeneous spaces are mathematical structures where each point looks the same as any other point from the perspective of the space's symmetries. This uniformity means that for any two points in the space, there is a symmetry transformation that moves one point to the other, which is particularly important in understanding geometric properties and structures like Riemannian manifolds, especially in the context of Riemannian submersions.
Hopf Fibration: The Hopf fibration is a specific type of mapping in topology that describes a way to project the 3-dimensional sphere, S^3, onto the 2-dimensional sphere, S^2, such that each point on S^2 corresponds to a circle in S^3. This fascinating structure reveals deep connections between different areas of mathematics, such as topology and differential geometry, and serves as a classic example of a Riemannian submersion, where fibers are embedded submanifolds in a higher-dimensional space.
Horizontal distribution: Horizontal distribution refers to a specific type of geometric structure in the context of Riemannian submersions, where the tangent space at each point of the total manifold can be decomposed into a direct sum of vertical and horizontal spaces. The horizontal space provides a way to project the geometry of the total manifold onto the base manifold, preserving certain geometric properties while allowing for the exploration of the relationship between these two spaces.
Kaluza-Klein Theory: Kaluza-Klein Theory is a theoretical framework that unifies general relativity and electromagnetism by introducing additional spatial dimensions beyond the familiar three. By compactifying these extra dimensions, it allows for a unified description of gravity and electromagnetism, suggesting that the fundamental forces may be manifestations of higher-dimensional spaces.
Local trivialization: Local trivialization refers to a property of a fiber bundle where, in a neighborhood of each point in the base space, the bundle looks like a product space. This means that locally, the fiber bundle can be represented as the Cartesian product of the base space and the fiber. This concept is crucial for understanding how geometric structures behave in smaller, manageable pieces, making it easier to analyze and work with complex structures.
O'Neill's Tensors: O'Neill's tensors are mathematical constructs used to analyze Riemannian submersions, providing a framework to understand the relationship between the geometry of a total space and its base space. They help capture how curvature and geometric properties transfer from one manifold to another, especially in the context of projections onto lower-dimensional spaces. This is particularly useful in studying the behavior of geometric structures under submersions, connecting various aspects of differential geometry.
Principal bundle: A principal bundle is a mathematical structure that formalizes the idea of having a space (the total space) that locally looks like a product of a base space and a group manifold. It consists of a total space, a base space, and a structure group, where the fibers over each point in the base space are homeomorphic to the group. This concept is essential for understanding various geometric structures, including Riemannian submersions and gauge theories, as it helps in organizing how different spaces relate to each other through symmetry transformations.
Pullback metric: A pullback metric is a way to define a Riemannian metric on a manifold by pulling back a metric from another manifold via a smooth map. This concept is crucial in understanding how geometric structures can be transferred between manifolds, particularly in the context of Riemannian submersions where one manifold projects onto another. The pullback metric allows us to study the properties of the original manifold through the lens of the structure on the target manifold.
Quotient Manifolds: Quotient manifolds are a way to construct new manifolds by taking an existing manifold and identifying certain points according to an equivalence relation. This process creates a new topological space that retains many properties of the original manifold while allowing for simplifications, such as reducing dimensions or analyzing symmetry. They play a crucial role in understanding various geometric structures, particularly in contexts like Riemannian submersions, where the geometry of the total space is projected onto a lower-dimensional base space.
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.
Riemannian submersion: A Riemannian submersion is a smooth surjective map between two Riemannian manifolds that preserves the length of tangent vectors from the manifold at the base to the total space, while projecting vertical vectors orthogonally. This concept is crucial as it relates to how geometrical structures can be preserved under mappings, and it connects with the study of curvature and parallel transport in differential geometry.
Smooth Map: A smooth map is a function between two smooth manifolds that preserves the structure of the manifolds by ensuring that the map is infinitely differentiable. This means that when you take derivatives of the map, all of them exist, making it crucial in studying the relationships and properties of smooth manifolds. Smooth maps are foundational in understanding how different geometric structures interact and play a vital role in various advanced topics like submersions and embeddings.
Submersion: Submersion is a smooth and surjective differential map between differentiable manifolds, where the differential at each point is surjective. This concept is vital in understanding how one manifold can be 'mapped down' onto another, preserving certain geometric structures. Submersions are particularly important in the context of studying Riemannian submersions and the behavior of embedded and immersed submanifolds, as they provide insights into how different geometric properties interact when transitioning from one manifold to another.
Total Space: In differential geometry, the total space refers to the complete manifold that serves as the domain for a given geometric structure, particularly in the context of fiber bundles, warped products, and Riemannian submersions. It is the setting where additional structures can be defined, such as fibers or metrics, influencing the way spaces relate to each other. Understanding the total space is crucial for analyzing properties like curvature and topological features across different dimensions and geometric configurations.
Totally geodesic fibers: Totally geodesic fibers refer to the fibers of a Riemannian submersion that are geodesics in the total space. This means that any curve in the fiber, when projected down to the base space, preserves its length and remains as a geodesic. In the context of Riemannian submersions, this property ensures that the geometry of the total space is well-preserved in the direction of the fibers.
Vertical Space: Vertical space refers to the tangent space associated with the fibers of a Riemannian submersion, which captures the notion of distance and angles in a way that reflects the geometry of the manifold. In this context, vertical space provides a geometric structure to study how points are related through projections onto lower-dimensional manifolds. Understanding vertical space is essential for analyzing how distances and curvature behave under these projections.
Warped Product Metrics: Warped product metrics are a way to construct new Riemannian manifolds by combining two different Riemannian manifolds in a specific way. This involves taking a base manifold and a fiber manifold, where the geometry of the fiber can vary over points in the base, allowing for richer geometric structures. This concept connects closely with Riemannian submersions, as warped products can be seen as a specific type of Riemannian submersion where the fibers are scaled differently at each point in the base manifold.
Yang-Mills Theory: Yang-Mills theory is a framework in theoretical physics that generalizes classical electromagnetism to non-abelian gauge groups, which are crucial in describing fundamental forces like the strong and weak nuclear interactions. It involves fields that take values in a Lie group, leading to gauge invariance, and connects closely with fiber bundles, where the geometry of the underlying space is enriched by the structure of the gauge group.