Editing The Classical Theory of Fields (Book)
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Kobayashi & Nomizu's book series is like a sequel to Sternberg's lectures in that its focus is on the definitions of geometric structures rather than algebraic invariants, and begins on connection theory where Sternberg's book ends. It also describes characteristic classes in an alternate way, via the Chern-Weil homomorphism with its starting point being curvature and connections rather than abstract algebraic invariants (since Milnor only references the curvature story in his appendix). Here, we can also get an impression of how gauge theory began in mathematics independently of its current incarnation in particle physics with Ehresmann, Weyl, and Cartan. Lie groups are the simplest curved spaces without the unnecessary restriction to two dimensions (there are no nonabelian 2-dimensional Lie groups), and their quotients offer alternative definitions of important topological spaces such as spheres, Grassmannians, Euclidean and Hyperbolic spaces. Continuing with Sternberg's explanation of the structure of Euclidean spaces, all of these 'symmetric spaces' and homogeneous spaces come with some naturally-equipped structures such as invariant 1-forms or metrics which are generalized to other compact Lie groups and used to give the most concrete constructions of curvature or other functions available in differential geometry. These spaces then became the starting point for more general bundle theory, as they are also equipped with natural bundles. Unlike these spaces, curvature in GR is not generally uniform and dynamically varies with the matter content of space-time. As such, Gauge theory is also equipped with the language to describe more general curvatures on manifolds. | Kobayashi & Nomizu's book series is like a sequel to Sternberg's lectures in that its focus is on the definitions of geometric structures rather than algebraic invariants, and begins on connection theory where Sternberg's book ends. It also describes characteristic classes in an alternate way, via the Chern-Weil homomorphism with its starting point being curvature and connections rather than abstract algebraic invariants (since Milnor only references the curvature story in his appendix). Here, we can also get an impression of how gauge theory began in mathematics independently of its current incarnation in particle physics with Ehresmann, Weyl, and Cartan. Lie groups are the simplest curved spaces without the unnecessary restriction to two dimensions (there are no nonabelian 2-dimensional Lie groups), and their quotients offer alternative definitions of important topological spaces such as spheres, Grassmannians, Euclidean and Hyperbolic spaces. Continuing with Sternberg's explanation of the structure of Euclidean spaces, all of these 'symmetric spaces' and homogeneous spaces come with some naturally-equipped structures such as invariant 1-forms or metrics which are generalized to other compact Lie groups and used to give the most concrete constructions of curvature or other functions available in differential geometry. These spaces then became the starting point for more general bundle theory, as they are also equipped with natural bundles. Unlike these spaces, curvature in GR is not generally uniform and dynamically varies with the matter content of space-time. As such, Gauge theory is also equipped with the language to describe more general curvatures on manifolds. | ||
Spin geometry specializes back to Riemannian geometry | Spin geometry specializes back to Riemannian geometry. | ||
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=== Applications === | === Applications === |