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Erschienen in:
2015 | OriginalPaper | Buchkapitel
Preliminaries
verfasst von : Paul Steinmann
Erschienen in: Geometrical Foundations of Continuum Mechanics
Verlag: Springer Berlin Heidelberg
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The treatment of geometry goes back to the ancient Greek, thereby the findings documented in the 13 volumes of Euclid’s Elements defined the state of affairs for some 2000 years. The advent of differential geometry is associated with the Habilitation lecture of Riemann in 1854. Its further development enabled and cumulated in the formulation of Einstein’s Theory of General Relativity/Gravitation some sixty years later in 1915. However the necessity for a non-Euclidean geometry may be motivated already from simply considering the failure of some of the corner stones in Euclidean geometry, for example the parallel axiom, on a two-dimensional curved manifold such as a sphere. Differential manifolds may be classified in terms of the two fundamental quantities connection and metric that in turn give rise to the three most essential tensors in differential geometry describing torsion, curvature, and non-metricity.