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Absolute Differential Calculus

In: The Physical and Mathematical Foundations of the Theory of Relativity

Author

Listed:
  • Antonio Romano

    (Università degli Studi di Napoli Federico II, Dipartimento di Matematica e Applicazioni “Renato Caccioppoli”)

  • Mario Mango Furnari

    (Istituto di Cibernetica)

Abstract

In this chapter, we address the fundamental problem of extending the differential calculus to manifolds. This extension requires the introduction of a criterion to compare vectors belonging to different tangent spaces of the manifold. This criterion, except for some general rules that it must satisfy, is quite arbitrary. A choice of the criterion corresponds to the introduction of an affine connection on the manifold. In this chapter we define the affine connections and study their properties. On a Riemannian manifold it is possible to define only one affine connection that is compatible with the metric.

Suggested Citation

  • Antonio Romano & Mario Mango Furnari, 2019. "Absolute Differential Calculus," Springer Books, in: The Physical and Mathematical Foundations of the Theory of Relativity, chapter 0, pages 103-128, Springer.
    • Handle: RePEc:spr:sprchp:978-3-030-27237-1_4
    DOI: 10.1007/978-3-030-27237-1_4
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