By Aurel Bejancu
This self-contained e-book starts off with the fundamental fabric on distributions and foliations. It then progressively introduces and builds the instruments wanted for learning the geometry of foliated manifolds. the most subject of the e-book is to enquire the interrelations among foliations of a manifold at the one hand, and the numerous geometric constructions that the manifold may possibly admit nevertheless. between those buildings are: affine, Riemannian, semi-Riemannian, Finsler, symplectic, advanced and get in touch with constructions. utilizing those constructions, the publication provides attention-grabbing periods of foliations whose geometry is especially wealthy and promising. those comprise the sessions of: Riemannian, completely geodesic, absolutely umbilical, minimum, parallel non-degenerate, parallel absolutely - null, parallel in part - null, symmetric, transversally symmetric, Lagrange, absolutely genuine and Legendre foliations. a few of these sessions look for the 1st time within the literature in booklet shape. ultimately, the vertical foliation of a vector package deal is used to strengthen a gauge concept at the overall area of a vector bundle.
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Extra resources for Foliations and Geometric Structures
3. Let (M, g) be a 2–dimensional Lorentz manifold and D be a totally–null line ﬁeld on M . 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.
7. The adapted linear connection determined by the pair of induced connections (∇, ∇⊥ ) is just the Schouten–Van Kampen connection ∇◦ deﬁned by the Levi–Civita connection ∇ on (M, g). 30 1 GEOMETRY OF DISTRIBUTIONS ON A MANIFOLD Proof. 19). Now we deﬁne two classes of manifolds that are going to be studied in detail in Chapter 4. When both distributions D and D⊥ are integrable, we say that M is a locally product manifold. If moreover, the local leaves of D and D⊥ are totally geodesic immersed in (M, g) then we say that M is a locally semi–Riemannian product.
32) AX Y = h (Q X, Q Y ) + h(Q X, QY ). 33) and obtain TX Y + AX Y = h(X, QY ) + h (X, Q Y ). 35) which relates the Levi–Civita and Schouten–Van Kampen connections on (M, g) via the tensor ﬁelds T and A. 16) becomes ∇◦X Y = ∇∗X Y + AQ Y QX + TQY Q X. 36) Therefore all the relations between curvature tensor ﬁelds of the linear connections we deﬁned on M , D and D⊥ can be expressed in terms of T and A. 18) into such a formula. 35) we deduce that ∇X ∇Y Z = ∇◦X ∇◦Y Z + ∇◦X (TY Z) + ∇◦X (AY Z) + TX (∇◦Y Z) +AX (∇◦Y Z) + TX (TY Z) + AX (TY Z) + TX (AY Z) + AX (AY Z).