Abstract
In this chapter, another fundamental concept is introduced, that of a vector bundle. The structure group of a vector bundle is a Lie group, and we shall therefore use this opportunity to also discuss Lie groups and their infinitesimal versions, the Lie algebras. Complex and symplectic structures are also discussed. Spin geometry is treated in detail.
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To be precise, Dirac [112] took the operator \(-\Delta \). This case will be addressed in an exercise.
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Every rotation of a plane is a product of two reflections, and the normal form of an orthogonal matrix shows that it can be represented as a product of rotations and reflections in mutually orthogonal planes.
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For the sake of the present discussion, we identify V with \(\mathbb{R}^{n}\) (\(n =\dim _{\mathbb{R}}V\)).
- 5.
The genus is a basic topological invariant of a compact surface. There are several different ways of defining or characterizing it, see [243]. For instance, it equals the first Betti number b 1, the dimension of the first cohomology, that will be defined in the next chapter.
- 6.
In the bibliography, a superscript will indicate the edition of a monograph. For instance,72017 means 7th edition, 2017.
Bibliography
In the bibliography, a superscript will indicate the edition of a monograph. For instance,72017 means 7th edition, 2017.
V.I. Arnold. Mathematical methods of classical mechanics. Springer-Verlag, Berlin, 1978.
M. Atiyah, R. Bott, and A. Shapiro. Clifford modules. Topology, 3(Suppl. I):3–38, 1964.
N. Berline, E. Getzler, and M. Vergne. Heat kernels and Dirac operators. Springer, 1992.
P. Dirac. The quantum theory of the electron. Proc. R. Soc. Lond. Ser. A, 778:618–624, 1928.
M. Gromov. Pseudoholomorphic curves in symplectic geometry. Inv. math., 82:307–347, 1985.
S. Helgason. Differential Geometry, Lie Groups and Symmetric Spaces. Academic Press, 1978.
H. Hofer and E. Zehnder. Symplectic invariants and Hamiltonian dynamics. Birkhäuser, 1994.
D. Husemoller. Fibre bundles. Springer, GTM 20, 1975.
J. Jost. Compact Riemann surfaces. Springer,32006.
J. Jost and X. Li-Jost. Calculus of variations. Cambridge Univ. Press, 1998.
H.B. Lawson and M.L. Michelsohn. Spin geometry. Princeton University Press, Princeton, 1989.
D. McDuff and D. Salamon. Introduction to symplectic topology. Oxford Univ.Press, 1995.
J. Morgan. The Seiberg-Witten equations and applications to the topology of smooth four manifolds. Princeton University Press, 1996.
J. Moser. On the volume elements of a manifold. Trans.AMS, 120:286–294, 1965.
A. Newlander and L. Nirenberg. Complex analytic coordinates in almost complex manifolds. Ann. Math., 65:391–404, 1957.
D. Salamon. Spin geometry and Seiberg-Witten invariants. 1995.
H. Wu. The Bochner technique in differential geometry. Math. Rep., 3(2):289–538, 1988.
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Jost, J. (2017). Chapter 2 Lie Groups and Vector Bundles. In: Riemannian Geometry and Geometric Analysis. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-319-61860-9_2
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DOI: https://doi.org/10.1007/978-3-319-61860-9_2
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