Abstract
An elementary introduction to the tensor representations of the diffeomorphism group and to the related covariant differential calculus. Includes, in particular, tensor densities, isometries, and a discussion of the properties of the affine connection. Of special interest (not easily available in the textbook literature): the explicit form of the contraction rules for totally antisymmetric tensors; the definition of infinitesimal transformations up to second order; the computation of the full affine connection including torsion and non-metricity contributions.
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Notes
- 1.
Differentiable manifold: a topological Hausdorff space locally homeomorphic to ℝn.
- 2.
When this condition is satisfied one also says that the metric has the property of “form-invariance”.
- 3.
This means that the functional variation we have considered can also be interpreted, geometrically, as the effect of an infinitesimal translation along the curve with parametric equation x μ=x μ(λ) and tangent vector ξ μ=dx μ/dλ.
References
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Weinberg, S.: Cosmology. Oxford University Press, Oxford (2008)
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Gasperini, M. (2013). Tensor Calculus in a Riemann Manifold. In: Theory of Gravitational Interactions. Undergraduate Lecture Notes in Physics. Springer, Milano. https://doi.org/10.1007/978-88-470-2691-9_3
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DOI: https://doi.org/10.1007/978-88-470-2691-9_3
Publisher Name: Springer, Milano
Print ISBN: 978-88-470-2690-2
Online ISBN: 978-88-470-2691-9
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