Absolute Differential Calculus

Antonio Romano () and Mario Mango Furnari ()
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Antonio Romano: Università degli Studi di Napoli Federico II, Dipartimento di Matematica e Applicazioni “Renato Caccioppoli”
Mario Mango Furnari: Consiglio Nazionale delle Ricerche, Istituto di Cibernetica

Chapter 4 in The Physical and Mathematical Foundations of the Theory of Relativity, 2026, pp 131-160 from Springer

Abstract: 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. This means that the connection does not modify the scalar product between two vectors when they undergo a parallel transport along an arbitrary curve, provided that the parallelism is evaluated by the affine connection.

Date: 2026
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DOI: 10.1007/978-3-032-23179-6_4

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