Associated fibre bundles - Lec 20 - Frederic Schuller
TLDRThe video script delves into the mathematical concept of associated bundles in the context of principal fiber bundles, which are significant in physics. It begins by defining associated bundles, which are constructed from a given principal bundle with a specific left action on a chosen smooth manifold. The script then illustrates the process of defining an equivalence relation to construct the associated bundle's total space and projection map, emphasizing their dependence on the principal bundle, smooth manifold, and left action. The discussion progresses to examples, starting with the frame bundle and its associated tangent bundle, highlighting the role of the general linear group and its action in defining the transformation properties of the associated bundle's components. The script further explores the construction of tensor bundles and introduces the concept of tensor densities, which are essential in integrating functions over manifolds. The video also explains the distinction between sections of an associated bundle and functions on a bundle, establishing a one-to-one correspondence between them. It concludes with the theorem on the restriction and extension of bundles, explaining the conditions under which a principal bundle can be restricted to a subgroup or extended from a closed subgroup, and its implications for manifolds equipped with various metrics. Overall, the script provides a comprehensive understanding of associated bundles, their construction, and their applications in physics and mathematics, particularly in the study of manifolds and their geometric structures.
Takeaways
- ๐ The concept of associated bundles is introduced as a way to connect principal fiber bundles to physical situations, providing more structure and insight into their importance in physics.
- ๐ Associated bundles are constructed from a given principal bundle with a specific left action on a new smooth manifold F, which is different from the right action on the principal bundle.
- ๐๏ธ The total space of an associated bundle, denoted as P_F, is created by taking the Cartesian product of the principal bundle's total space and the new manifold F, then identifying elements under an equivalence relation.
- ๐ The associated bundle has a projection map, ฯ_F, which is well-defined and projects the associated bundle onto the same base space M as the principal bundle.
- ๐ The construction of the total space P_F and the projection map ฯ_F results in a fiber bundle with a typical fiber F, which is tied to the principal bundle through the action.
- ๐ผ๏ธ The script provides an example using the frame bundle, which is a principal bundle where the fibers are bases of the tangent space at each point of the manifold.
- โ๏ธ A left action of the general linear group GL(D, R) on the fiber F is defined, which is essential for constructing the associated bundle from the frame bundle.
- ๐ The tangent bundle is shown to be isomorphic to an associated bundle constructed from the frame bundle, with the isomorphism provided by a specific map that relates elements of the associated bundle to tangent vectors.
- ๐ค The script discusses the implications of choosing different groups for the action on the fiber, such as the possibility of restricting to the Lorentz group for applications in special relativity.
- ๐ The construction of tensor bundles and tensor density bundles is explained, highlighting the role of the determinant of the group element in defining the action and the resulting bundle.
- ๐งฎ The importance of tensor densities in integration over manifolds is emphasized, as they account for changes in the volume element under changes of coordinates.
- โ๏ธ The script touches on the concept of bundle maps between associated bundles, which are constructed from principal bundle maps, and the distinction between trivial fiber bundles and trivial associated bundles.
Q & A
What is an associated bundle in the context of principal fiber bundles?
-An associated bundle is a bundle that is associated with a principal fiber bundle in a precise sense. It is constructed using a principal bundle, a smooth manifold F, and a left G-action on F, which is tied to the principal bundle through the action.
How is the total space of an associated bundle constructed?
-The total space of an associated bundle, denoted as PF, is constructed by taking the Cartesian product of the total space of the principal bundle P and the manifold F, and then defining an equivalence relation on P ร F that identifies elements based on the right action of G on P and the left action of G on F.
What is the significance of the left G-action in defining an associated bundle?
-The left G-action on the smooth manifold F is significant because it allows the construction of the associated bundle to be dependent on the choice of the group G and the action of this group on the fiber F. This action is what differentiates the associated bundle from the principal bundle.
How does the projection map of an associated bundle relate to the projection map of the principal bundle?
-The projection map of an associated bundle, denoted as ฯF, is defined such that for an element of the associated bundle (an equivalence class of a pair from the principal bundle and the fiber), it sends this element to the point in the base space M that is obtained by applying the projection map ฯ from the principal bundle to the principal bundle element of the pair.
What is the relationship between the tangent bundle and the frame bundle?
-The tangent bundle can be considered as an associated bundle to the frame bundle. The frame bundle provides a local frame for the tangent space at each point of the manifold, and the tangent bundle is constructed by applying a left action of the general linear group GL(D, R) on the fiber, which is the vector space R^D.
What is the role of the group GL(D, R) in the construction of the tangent bundle as an associated bundle?
-The group GL(D, R) plays a crucial role in defining how the components of a vector in the tangent space transform under a change of frame. This transformation property is essential for the tangent bundle to be an associated bundle to the frame bundle.
How does the concept of tensor densities relate to the construction of associated bundles?
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What is the importance of the theorem stating that sections of an associated bundle are in one-to-one correspondence with F-valued functions on the underlying principal bundle?
-This theorem provides a way to understand sections of associated bundles in terms of functions on the underlying principal bundle. It offers an alternative perspective on vector fields, for instance, by allowing them to be viewed as F-valued functions on the frame bundle.
What does it mean for an associated bundle to be trivial?
-An associated bundle is considered trivial if the underlying principal bundle is trivial. This means there exists a bundle isomorphism between the associated bundle and the product of its base space and fiber. Triviality of an associated bundle implies a specific structure that allows for a global section that is invariant under the group action.
What is the condition for a principal G-bundle to be restricted to an H-principal bundle, where H is a closed subgroup of G?
-A principal G-bundle can be restricted to an H-principal bundle if H is a closed subgroup of G and there exists a bundle map that respects the right action of H on the total space of the principal bundle.
How does the concept of a cross-section relate to the triviality of a principal bundle?
-A principal bundle is considered trivial if and only if it has a cross-section. A cross-section is a map that assigns to each point in the base space an element in the total space of the bundle in a way that is consistent with the bundle's structure.
What is the significance of the theorem regarding the restriction and extension of bundles in the context of physics?
-The theorem regarding the restriction and extension of bundles is significant in physics because it provides a mathematical framework for understanding how physical fields and particles transform under different symmetry groups. This is particularly relevant in the theory of relativity and quantum mechanics, where the choice of symmetry group can have profound implications for the physical interpretation of the theory.
Outlines
๐ Introduction to Associated Bundles
The video begins with an introduction to associated bundles, which are mathematical constructs that provide a deeper understanding of principal fiber bundles. These are discussed in the context of physics, emphasizing their significance in the field. The concept of a principal bundle is first reviewed, followed by the precise definition of an associated bundle, which involves a G-principal bundle, a smooth manifold F, and a left G-action on F. The construction of the associated bundle involves an equivalence relation and the creation of a new bundle space P_F with a projection map ฯ_F.
๐ Equivalence Relation and Quotient Space in Associated Bundles
The script explains the process of defining an equivalence relation on the Cartesian product of the principal bundle's total space and the manifold F. This relation identifies elements based on the action of the group G. The quotient space formed by this relation is denoted as P_F and is considered to be the total space of a new bundle with the base space M. The fibers of this bundle are tied to the principal bundle through the action of G.
๐ Well-Definedness of the Associated Projection Map
The paragraph discusses the construction of the associated projection map ฯ_F, which is well-defined due to the properties of the equivalence relation. The map takes an element of the associated bundle, viewed as an equivalence class, and projects it to a point in M using the principal bundle's projection map ฯ. The script also addresses the need to check the map's well-definedness by considering different representatives of the equivalence class.
๐ Examples of Associated Bundles: Frame Bundle and Tangent Bundle
The video provides examples of associated bundles, starting with the frame bundle, which is a principal bundle where each point of the manifold has a basis of the tangent space. The script then connects this to the concept of the tangent bundle as an associated bundle with the fiber being R^D. The left action of GL(D, R) on the fiber is defined, and the resulting associated bundle is shown to be isomorphic to the tangent bundle.
๐ Transformation of Components in Associated Bundles
The script explains how the components of elements in the tangent space transform when the frame is changed by a general linear transformation. This transformation is imposed when constructing an associated bundle, and the script emphasizes that this choice of transformation is what physicists often refer to when discussing how components change under a frame change.
โ๏ธ Bundle Isomorphism between Constructed Bundle and Tangent Bundle
The video claims the existence of a bundle isomorphism between the newly constructed associated bundle and the known tangent bundle. This isomorphism is defined by a map that takes an element of the associated bundle and sends it to an element of the tangent bundle. The components of the fiber element transform in the opposite direction under the group action, which is a key concept in physics.
The paragraph discusses the construction of a map that is independent of the choice of frame. It emphasizes that the map is invertible and that the choice of frame does not affect the result due to the equivalence class defined by the relation. The script also touches on the concept of tensors and their transformation properties under different groups, such as GLD and Lorentz transformations.
๐ Tensor Densities and Their Transformation Laws
The script introduces the concept of tensor densities, which are defined by modifying the left action of the group on the fiber to include an additional factor, the determinant of the group element's inverse raised to a power Omega. This modification results in a new type of bundle called the tensor density bundle. The script also differentiates between tensor bundles and tensor density bundles.
๐งฎ Scalar Densities and Their Significance
The video defines scalar densities as a special case of tensor densities where the fiber is kept invariant and only scaled by the determinant of the group element's inverse. The script explains that scalar densities are important in situations where integration over a manifold is required, as they allow for well-defined integrals under changes of coordinates.
๐ฌ General Relativity and the Role of Densities
The script connects the concept of densities to general relativity, where the metric tensor transforms as a density under general linear group transformations. This transformation property is crucial for understanding the Einstein action and the integration of Lagrangian densities in the theory.
๐ Associated Bundle Maps and Their Construction
The video defines associated bundle maps, which are bundle maps constructed from a principal bundle map between the underlying principal bundles. The script explains the conditions for such a map to exist and how it accounts for the specific structure of associated bundles.
โ๏ธ Triviality of Associated Bundles
The script discusses the concept of triviality for associated bundles, explaining that an associated bundle is considered trivial if the underlying principal bundle is trivial. The script also highlights the difference between a trivial fiber bundle and a trivial associated bundle, noting that the latter carries more structure.
๐ Restriction and Extension of Bundles
The video introduces the idea of restricting and extending bundles with respect to different underlying principal bundles. The script defines when a principal bundle can be restricted to a subgroup or extended from a closed subgroup and discusses the conditions under which these operations are possible.
๐ Cross Sections and the Possibility of Restriction
The script presents a theorem that determines whether a principal bundle can be restricted to a subgroup based on the existence of a cross section for the quotient bundle. The video provides examples related to the frame bundles and their ability to be equipped with different metrics, such as Riemannian or Lorentzian, and discusses the implications for the structure of the manifold.
๐ซ Limitations on Lorentzian Manifolds and Spin Structures
The video concludes with a discussion on the limitations of Lorentzian manifolds and the conditions under which they can support spin structures. The script highlights that not all manifolds can carry a spin structure, even if they admit a Lorentzian metric, which is an important consideration in the study of fermions in the context of general relativity.
Mindmap
Keywords
๐กAssociated Bundles
๐กPrincipal Fiber Bundle
๐กGroup Action
๐กEquivalence Relation
๐กTangent Bundle
๐กFrame Bundle
๐กTensor Bundle
๐กTensor Density
๐กScalar Density
๐กBundle Map
๐กTrivial Bundle
Highlights
Introduction to associated bundles as a key concept in understanding the importance of principal fiber bundles in physics.
Definition of an associated bundle using a G principal bundle, a smooth manifold F, and a left G action on F.
Explanation of the construction of the associated bundle's total space PF and the projection map ฯF.
Discussion on how associated bundles allow for a deeper understanding of the relationship between physics and mathematics.
Use of the frame bundle as an example to illustrate the concept of associated bundles in the context of physics.
Construction of the tangent bundle as an associated bundle to the frame bundle, providing a new perspective on the tangent bundle.
Introduction of the concept of tensor densities as a new type of associated bundle, different from the tensor bundles.
Explanation of how the action of the group GLD on the fiber results in tensor density bundles, which are important in physics.
Discussion on the transformation properties of tensor densities under changes of frames in the principal bundle.
The importance of the determinant of the group element in defining the action on the fiber for tensor densities.
Differentiation between scalar densities and tensor densities based on the weight factor in their transformation properties.
Insight into how densities are crucial when integrating over manifolds, especially in the context of general relativity.
Clarification on the distinction between a section of a bundle and an F-valued function on the underlying principal bundle.
Theorem stating that sections of an associated bundle correspond one-to-one with F-valued functions on the principal bundle.
Exploration of the conditions under which a principal bundle can be trivial, and the implications for associated bundles.
Definition of bundle maps between associated bundles and how they are induced from principal bundle maps.
Discussion on the extension and restriction of bundles with respect to different underlying principal bundles and their groups.
The significance of the theorem that determines when a principal bundle can be restricted to a subgroup or extended from a subgroup.
Implications of the theorem for the study of manifolds equipped with various metrics, such as Riemannian or Lorentzian, and their relation to spin structures.
Transcripts
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