By Aurel Bejancu
This self-contained publication begins with the elemental fabric on distributions and foliations. It then progressively introduces and builds the instruments wanted for learning the geometry of foliated manifolds. the most subject matter of the e-book is to enquire the interrelations among foliations of a manifold at the one hand, and the various geometric constructions that the manifold may well admit however. between those constructions are: affine, Riemannian, semi-Riemannian, Finsler, symplectic, complicated and make contact with constructions. utilizing those buildings, the e-book provides fascinating periods of foliations whose geometry is particularly wealthy and promising. those comprise the periods of: Riemannian, completely geodesic, absolutely umbilical, minimum, parallel non-degenerate, parallel completely - null, parallel partly - null, symmetric, transversally symmetric, Lagrange, completely actual and Legendre foliations. a few of these sessions look for the 1st time within the literature in e-book shape. eventually, the vertical foliation of a vector package is used to advance a gauge thought at the overall area of a vector bundle.
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Extra info for Foliations and Geometric Structures
24 1 GEOMETRY OF DISTRIBUTIONS ON A MANIFOLD Proof. 4) for any X, Y, Z ∈ Γ (T M ). 4) is a linear connection on D that satisﬁes the conditions (i) and (ii). Next, suppose that ∇ is another linear connection on D satisfying (i) and (ii). 6) and for any X, Y ∈ Γ (T M ). 7) − 2g(∇QX QY, QZ). 3) respectively, we conclude that ∇ = D, which proves the uniqueness of D. 14)) ∇X Y − ∇Y X − [X, Y ] = 0, ∀ X, Y ∈ Γ (T M ). 8) If (M, g) is a semi–Riemannian manifold then we say that g is parallel with respect to ∇ if we have (∇X g)(Y, Z) = X(g(Y, Z)) − g(∇X Y, Z) − g(Y, ∇X Z) = 0, ∀ X, Y, Z ∈ Γ (T M ).
Iv) ∇∗ = ∇. 10 we see that the condition ∇ = D on D is stronger than the condition for g being parallel with respect to D. The latter condition was ﬁrst introduced by Reinhart [Rei59a] for foliated manifolds, that is, D⊥ is supposed to be an integrable distribution. 3 for more details). It is interesting to see whether bundle–like metrics can be found on a non–holonomic semi–Riemannian manifold (M, g, D, D⊥ ), that is, when none of the distributions D and D⊥ is integrable. The next example shows that the answer is in the aﬃrmative.
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). 38) + A∇◦X Y Z − A∇◦Y X Z − AT ◦ (X,Y ) Z. 39) 40 1 GEOMETRY OF DISTRIBUTIONS ON A MANIFOLD and (∇◦X A)Y Z = ∇◦X (AY Z) − A∇◦X Y Z − AY (∇◦X Z). 18). 41). 43) for any X, Y, Z ∈ Γ (T M ), where the covariant derivative of B is given by (∇◦X B)(Y, Z) = ∇◦X (B(Y, Z)) − B(∇◦X Y, Z) − B(Y, ∇◦X Z). 44) we deduce that (∇◦X B)(Y, Z) = (∇⊥ X h)(Y, QZ) + (∇X h )(Y, Q Z) = (∇◦X T )Y Z + (∇◦X A)Y Z.
Foliations and Geometric Structures by Aurel Bejancu