Abstract
This Chapter introduces basic notions from differential geometry and classical field theories, including some less standard material, such as, e.g. boundaries and singularities. The Chapter begins with definitions from the Riemannian geometry with a focus on mathematical foundations of the general relativity theory. The material includes gravity actions, examples of dynamical equations and their physically important solutions. Description of isometries contains definitions of Killing vectors and Killing spinors. Characteristics of hypersurfaces are explained in detail to set a stage for theories with boundaries. Defects of geometry located on codimension one and codimension two hypersurfaces are considered as examples of singularities when the curvature behaves as a distribution. These two cases are among the base manifolds where a heat kernel operator is studied in next chapters. Discussion of field models starts with a brief introduction to fiber bundles and the structure groups including basic features of the spin group. Properties of the modern elementary particle physics are exposed by using models of dynamical scalar, spinor, massive vector, and gauge fields propagating in classical backgrounds.
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Fursaev, D., Vassilevich, D. (2011). Geometrical Background. In: Operators, Geometry and Quanta. Theoretical and Mathematical Physics. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-0205-9_1
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DOI: https://doi.org/10.1007/978-94-007-0205-9_1
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