By Aurel Bejancu
The ebook is self-contained. It starts off with the fundamental fabric on distributions and foliations. Then it proceeds to progressively introduce and construct the instruments wanted for learning the geometry of foliated manifolds. the most subject of the ebook is to enquire the interrelations among foliations of a manifold on one hand, and the various geometric buildings that the manifold may perhaps admit nevertheless. between those buildings we point out: affine, Riemannian, semi-Riemannian, Finsler, symplectic, advanced and get in touch with buildings. utilizing those constructions, the ebook provides fascinating periods of foliations whose geometry is especially wealthy and promising. those contain the periods of : Riemannian, completely geodesic, completely umbilical, minimum, parallel non-degenerate, parallel absolutely - null, parallel in part - null, symmetric, transversally symmetric, Lagrange, absolutely actual and Legendre foliations. a few of these sessions look for the 1st time within the literature in a e-book shape. eventually, the vertical foliation of a vector package is used to improve a gauge thought at the overall area of a vector bundle.
The booklet should be of curiosity to graduate scholars who are looking to be brought to the geometry of foliations, to researchers engaged on foliations and geometric buildings, and to physicists drawn to gauge theories.
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Additional resources for Foliations and Geometric Structures (Mathematics and Its Applications)
As in the case n = 1 we look for a totally–null transversal distribution to D. To this end, we consider a complementary distribution S to D⊥ in D, that is, we have D = S ⊕ D⊥ . 11) As any Sx is a screen subspace of Dx , we call S a screen distribution for D. 6 we deduce that S is a non–degenerate distribution on M . 3 we deduce that the complementary orthogonal distribution S ⊥ to S in T M is non–degenerate too. 12) and D⊥ is a vector subbundle of S ⊥ . Then we consider a complementary dis⊥ tribution H to D⊥ in S ⊥ .
Then we have the following assertions: (i) The Levi–Civita connection ∇ on (M, g) is an adapted connection on the almost product manifold (M, D, D ). (ii) The integral curves of both D and D are null pregeodesics of (M, g). Next, we consider the case n > 1. 1, D⊥ is a vector subbundle of D and therefore any complementary distribution to D in T M is not orthogonal to D. Moreover D is a partially–null distribution on M because it is a 1-degenerate n–distribution with n > 1. As in the case n = 1 we look for a totally–null transversal distribution to D.
6. Let (W, g) be a partially–null subspace of a semi–Euclidean space (V, g). Then any complementary subspace to N (W, g) in W is non–degenerate. Proof. Let S(W, g) be a complementary subspace to N (W, g) in W . Suppose that S(W, g) is degenerate. 4 From Semi–Euclidean Algebra to Semi–Riemannian Geometry 21 such that g(v, w) = 0 for any w ∈ S(W, g). As v ∈ W and N (W, g) is the null subspace of W , we have g(v, u) = 0, for any u ∈ N (W, g). Thus v is orthogonal to all vectors of W and hence it is a vector in N (W, g).