Differential Geometry

This chapter explores the intricate world of classical field theories defined on arbitrary manifolds, introducing the tools of differential geometry. The key concepts of connection and parallel transport of various mathematical objects, scalar, tensor, or spinor fields, are introduced. The discussion begins with an exploration of metric tensors, essential for understanding the geometric properties of spacetime. A detailed methodology for calculating Levi-Civita connection coefficients is presented, emphasizing the transformation of indices and the role of matrix reshaping. This process illuminates the relationship between torsion and curvature in the spacetime manifold. The derivation proceeds with the construction of Christoffel symbols, which are pivotal for understanding the geometric structure and the resulting gravitational effects. Spin connection coefficients are then introduced, showcasing their relevance in describing spacetime symmetries and the behaviour of spinors. The chapter culminates with formulating the Fock-Ivanenko connection, which is crucial for extending the framework to incorporate fermionic fields. Practical computational techniques and symbolic notations, such as Penrose graphical notation, are discussed, providing alternative perspectives on tensor analysis and enhancing comprehension of complex field interactions in gravitational physics.

Authors and Affiliations

1. Université de Lorraine, Nancy, France

   Bertrand Berche

2. Departamento de Fisica, Colegio de Ciencias e Ingeniería, Universidad San Francisco de Quito, Quito, Ecuador

   Ernesto Medina

Corresponding author

Correspondence to Bertrand Berche .

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