Abstract
In the present chapter we provide most of the prerequisites for reading the rest of the book. In the first two sections we present a review of vector bundles and introduce the main differential operators: Lie derivative, exterior differential, linear connection, general connection. Distributions on manifolds (known as non-holonomic spaces in classical terminology) are then introduced and studied by using both methods of vector fields and of differential 1-forms. We give here the characterization for the existence of a transversal distribution to a foliation, which is found to be very useful in Chapters 4 and 5 for a general study of lightlike submanifolds. In the last two sections we deal with semi-Riemannian manifolds and lightlike manifolds. While the geometry of a semi-Riemannian manifold is fully developed by using the Levi-Civita connection we stress the role of the radical distribution in studying the geometry of a lightlike manifold. The main formulas and results are expressed by using both the invariant form and the index form.
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© 1996 Springer Science+Business Media Dordrecht
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Duggal, K.L., Bejancu, A. (1996). Differential-Geometric Structures On Manifolds. In: Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications. Mathematics and Its Applications, vol 364. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-2089-2_2
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DOI: https://doi.org/10.1007/978-94-017-2089-2_2
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-4678-9
Online ISBN: 978-94-017-2089-2
eBook Packages: Springer Book Archive