Abstract
Submanifolds are common and unavoidable objects. While in ordinary geometry they are low-dimensional, in mechanics they can be significantly more complicated. For example, the space of configurations of a solid body with one fixed point can be identified with the group SO(3) of rotations of R3, which is a 3-dimensional submanifold of the space R9 of 3 × 3 matrices. The configuration space of a (free) solid has three more dimensions due to translations; its phase space, taking account of velocities, has dimension 12. Likewise the phase space for the three-body problem has dimension 3 × 3 × 2 = 18.
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Notes
To avoid unnecessary complications we shall in general assume that the maps we consider have maximal regularity. Thus in future we shall say “diffeomorphism” to mean “diffeomorphism of class C∞”. Most theorems stated in the C∞ setting remain true with weaker hypotheses.
In fact this definition implies that a submanifold is always a’ locally closed’ subset (see 3.2.2): it is an open subset of its closure, or equivalently the intersection of an open set and a closed set. Note that the definition says nothing about the appearance of a submanifold in the neighbourhood of a point that does not belong to it (see the example at the end of Section 2.10).
Hassler W(UPHITNEY), 1907-1989, taught at Harvard from 1930 to 1952 and then at the Institute for Advanced Study from 1952 to 1977. He was one of the founders of the theory of differentiable maps. Many fine and beautiful theorems are due to him, and we shall meet some of them in this text.
Take care not to confuse “not being a critical value” with “being a non-critical value”, cf. 3.6.6.
The Norwegian Niels A(UPBEL) (1802-1829) and the young Evariste GALOIS (18111832) born at Bourg-La-Reine near Paris had almost parallel lives and mathematical work. It was Abel who proved the insolubility by radicals of the general equation of the fifth degree; at the time of his death he had almost arrived at the general results that Galois was to obtain three years later. The latter in particular extended Abel’s theorem to every degree ≧5.
See for example [BL], p.25.
Carl Friedrich G(UPAUSS) (1777-1855), one of the giants in the history of science and called “the prince of mathematicians”: he was certainly one of the greatest, with Archimedes, Euler, Riemann and a few others including some living today. The (bilingual) text of his Disquisitiones Generales—not to be confused with the Dis-quisitiones Arithmeticas, which is equally famous and important—accompanied by a historical study by Peter Dombrowski can be found in [DO] which is strongly recommended.
Bernhard R(UPIEMANN) (1826-1866) was a German mathematician and Professor at Göttingen from 1854 until his death. He contributed decisively to all areas of mathematics. The subject for this inaugural lecture had been assigned to him by Gauss.
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Demazure, M. (2000). Submanifolds. In: Bifurcations and Catastrophes. Universitext. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-57134-3_3
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DOI: https://doi.org/10.1007/978-3-642-57134-3_3
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