Abstract
Let us recall the notion of abstract smooth manifolds, that are not necessarily subsets of some \(\mathbb{R}^{m}\). As a matter of fact manifolds are very often ‘embedded manifolds’, for example, in analytic mechanics, a system of n particles subject to holonomic constraints evolves inside a manifold which is intrinsically given as a subset of \(\mathbb{R}^{3n}\). On the other hand it happens that the configuration manifold of a rigid body, \(\mathbb{R}^{3} \times \mathit{SO}(3)\), is in no ‘natural’ way a subset1 of some \(\mathbb{R}^{m}\). Not only because of this example, but for the need of a general setup, we will introduce such abstract structures.
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Notes
- 1.
Chosen so that x = 0 represents q and starting form the identity: \(\int _{0}^{1} \frac{d} {\mathit{dt}}f(\mathit{tx})\mathit{dt} = f(x) - f(0)\).
- 2.
Note that the derivations are necessarily vanishing on the constant functions: v(α β) = α β v(1) and v(α β) = α v(β) + v(α)β = 2α β v(1).
- 3.
The chart \(\varphi\) represents the curve γ (i) by means of the i-th coordinate line of \(\mathbb{R}^{n}\) through q.
- 4.
That is, if \(f =\sum _{\mathit{ij}}f_{\mathit{ij}}e^{{\ast}i} \otimes e^{{\ast}j}\), then f ij = −f ji .
- 5.
That is, if \(f =\sum _{i_{1},\ldots,i_{k}}f_{i_{1},\ldots,i_{k}}e^{{\ast}i_{1}} \otimes \ldots \otimes e^{{\ast}i_{k}}\), then \(\forall \,i_{\alpha },i_{\beta } \in \{ 1,\ldots,n\}\), \(f_{i_{1},\ldots,i_{\alpha },\ldots,i_{\beta },\ldots,i_{k}} = -f_{i_{1},\ldots,i_{\beta },\ldots,i_{\alpha },\ldots,i_{k}}\).
- 6.
Or, shortly, Riemann metrics.
- 7.
- 8.
The formula (1.11) is true only if the metric is Riemannian, i.e. positive defined; in the relativistic case, there is a further sign minus, see next pages.
- 9.
Even though not quite the most general.
- 10.
It is the same of ∗∗α = (−1)1+p(n−p) α for n = 4.
- 11.
Note that it is a relation between 4-forms (or, ‘volume’ forms).
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Cardin, F. (2015). Notes on Differential Geometry. In: Elementary Symplectic Topology and Mechanics. Lecture Notes of the Unione Matematica Italiana, vol 16. Springer, Cham. https://doi.org/10.1007/978-3-319-11026-4_1
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