Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Bücher
Zeitschriften
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
SLX-Digitalkonferenz: Zukunftswerkstatt 2024

Mehr Umsatz mit Top-Kontakten

Am 7. Mai erfahren Sie, wie Vertriebsteams Leadgenerierung, Leadnurturing und Leadstrategien effizient steuern können.
Erweiterte Suche
Anmelden
nach oben
Erschienen in:
Buchtitelbild

2015 | OriginalPaper | Buchkapitel

1. Notes on Differential Geometry

verfasst von : Franco Cardin

Erschienen in: Elementary Symplectic Topology and Mechanics

Verlag: Springer International Publishing

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

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.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Jetzt informieren
Nächstes Kapitel Symplectic Manifolds
Fußnoten
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
This definition is in agreement with [97], but not e.g. with [46].
 
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).
 
Literatur
2.
R. Abraham, J.E. Marsden, Foundations of Mechanics. Advanced Book Program, 2nd edn. (Benjamin/Cummings Publishing, Reading, 1978), xxii+m–xvi+806pp
7.
V.I. Arnol’d, Mathematical Methods of Classical Mechanics, 2nd edn. (Springer, New York, 1989)
8.
V.I. Arnol’d, A. Avez, Ergodic Problems of Classical Mechanics, Translated from the French by A. Avez (W. A. Benjamin, New York/Amsterdam, 1968), ix+286pp
36.
C. Chevalley, Theory of Lie Groups. I. Princeton Mathematical Series, vol. 8 (Princeton University Press, Princeton, 1946), ix+217pp
37.
R.L. Cohen, Immersions of manifolds. Proc. Natl. Acad. Sci. U.S.A. 79(10), 3390–3392 (1982)CrossRefMATH
42.
J. Dieudonné, Treatise on Analysis, Vol. III. Pure and Applied Mathematics, vol. 10-III (Academic, New York/London, 1972), xvii+388pp
44.
S.K. Donaldson, Self-dual connections and the topology of smooth 4-manifolds. Bull. Am. Math. Soc. (N.S.) 8(1), 81–83 (1983)
45.
B.A. Dubrovin, A.T. Fomenko, S.P. Novikov, Modern Geometry—Methods and Applications. Part II. The Geometry and Topology of Manifolds. Graduate Texts in Mathematics, vol. 104 (Springer, New York, 1985), xv+430pp
46.
B.A. Dubrovin, A.T. Fomenko, S.P. Novikov, Modern Geometry—Methods and Applications. Part III. Introduction to Homology Theory. Graduate Texts in Mathematics, vol. 124 (Springer, New York, 1990), x+416pp
47.
B.A. Dubrovin, A.T. Fomenko, S.P. Novikov, Modern Geometry—Methods and Applications. Part I. The Geometry of Surfaces, Transformation Groups, and Fields. Graduate Texts in Mathematics, vol. 93, 2nd edn. (Springer, New York, 1992), xvi+468pp
82.
T. Levi-Civita, The Absolute Differential Calculus (Translated from the Italian by Marjorie Long. Edited by Enrico Persico. Reprint of the 1926 translation. Dover Phoenix Editions) (Dover Publications, Mineola, 2005), xvi+452pp
83.
P. Libermann, C.-M. Marle, Symplectic Geometry and Analytical Mechanics. Mathematics and its Applications, vol. 35 (D. Reidel, Dordrecht, 1987), xvi+526pp
92.
J. Milnor, On manifolds homeomorphic to the 7-sphere. Ann. Math. (2) 64, 399–405 (1956)
94.
C.W. Misner, K.S. Thorne, J.A. Wheeler, Gravitation (WH Freeman, San Francisco, 1973)
97.
M. Nakahara, Geometry, Topology and Physics. Graduate Student Series in Physics (Institute of Physics Publishing, Bristol/Philadelphia, 1995)
98.
J. Nash, The imbedding problem for Riemannian manifolds. Ann. Math. (2) 63, 20–63 (1956)
99.
104.
F. Rampazzo, H.J. Sussmann, Set-valued differentials and a nonsmooth version of Chow’s theorem, in Proceedings of the 40th IEEE Conference on Decision and Control, vol. 3 (IEEE, New York, 2001), pp. 2613–2618
125.
H. Whitney, Analytic extensions of differentiable functions defined in closed sets. Trans. A. M. S. 36, 63–89 (1934); Differentiable functions defined in closed sets I. Trans. A. M. S. 36, 369–387 (1934); Functions differentiable on the boundaries of regions. Ann. Math. 35, 482–485 (1934); Differentiable manifolds, Ann. of Math. 37(2), no. 3, 645–680 (1936).
Metadaten
Titel
Notes on Differential Geometry
verfasst von
Franco Cardin
Copyright-Jahr
2015
Buch
Elementary Symplectic Topology and Mechanics

Print ISBN: 978-3-319-11025-7

Electronic ISBN: 978-3-319-11026-4

Copyright-Jahr: 2015

DOI
https://doi.org/10.1007/978-3-319-11026-4_1