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Mike2
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Is a vector bundle just a generalization of a Tangent bundle, where the vectors no longer have to be tangent to the manifold but can have components normal to it? Or is there more to it than that? Thanks.
A vector bundle is a mathematical structure that is used to describe spaces that vary smoothly in different directions. It is a collection of vector spaces that are smoothly connected over a given space, typically a topological space.
While both a vector bundle and a vector space are collections of vector spaces, a vector bundle is defined over a larger space and has a continuous structure. In contrast, a vector space is defined over a field and has a discrete structure.
A vector bundle has two main components: a base space and a vector space. The base space is the topological space over which the vector bundle is defined, and the vector space is the collection of vector spaces that vary smoothly over the base space.
Vector bundles have various applications in mathematics and physics. They are commonly used in differential geometry, topology, and algebraic geometry. In physics, vector bundles are used to describe physical quantities that vary smoothly over a given space, such as velocity or force fields.
Vector bundles are classified based on their topological properties, such as their dimension, orientation, and connectivity. This classification is known as characteristic classes and is used to distinguish different types of vector bundles.