By Aurelian Bejancu Hani Reda Farran A. Bejancu
This self-contained e-book begins 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 topic 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 nonetheless. between those structures are: affine, Riemannian, semi-Riemannian, Finsler, symplectic, complicated and phone buildings. utilizing those constructions, the booklet offers fascinating sessions of foliations whose geometry is particularly wealthy and promising. those comprise the sessions of: Riemannian, completely geodesic, absolutely umbilical, minimum, parallel non-degenerate, parallel completely - null, parallel partly - 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 is used to advance a gauge thought at the overall area of a vector package deal.
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14) for ∇◦ imply ∇[X,Y ] QZ = ∇[X,Y ] QZ + h(∇◦X Y, QZ) − h(∇◦Y X, QZ) − h(T ◦ (X, Y ), QZ). 9) ◦ − (∇⊥ Y h)(X, QZ) + h(T (X, Y ), QZ)}. 4) and g(R(X,Y )QZ,QU ) = g(R(X,Y )QZ,QU ) + g(h (X, h(Y,QZ)),QU ) − g(h (Y, h(X, QZ)), QU ). 3). 11) + h(X, h (Y, Q Z)) − h(Y, h (X, Q Z))}. 11) by taking the D⊥ – and D–components respectively. 6) the fundamental equations of the pair of distributions (D, D⊥ ) on (M, g). 6) are equivalent to each other. To see this we first prove the following lemma. 2. (i) The covariant derivatives of h and h are related by g((∇⊥ X h)(Y, QZ), Q U ) + g((∇X h )(Y, Q U ), QZ) = 0.
12) are not satisfied by R◦ . 16) but we can not claim that K ◦ determines R◦ on D. This is another reason for saying that the Vr˘ anceanu connection is more intimately related to the geometry of non–holonomic manifolds. 16) but using the curvature tensor field R of the Levi–Civita connection on (M, g) (see O’Neill [O83], p. 77). Then we can relate K, K ∗ and K ◦ as in the next theorem. 6. Let (M, g, D) be a semi–Riemannian non–holonomic manifold such that g is Vr˘ anceanu–parallel on D. 19) where {QX, QY } is an arbitrary basis of the non–degenerate D–plane W .
Hence [X,Y ]∈Γ / (D), which proves the assertion (i). 37)) becomes g([Z, X], Y ) + g([Z, Y ], X) = 0. This is a consequence of ∂ , ∂x1 [Z, X] = [Z, Y ] = −kf (t)f (t) and g ∂ , Y ∂x1 +g ∂ , X ∂x1 = 0. Thus the proof is complete. 28) ∇∗Y X = ∇∗X X = kf (1 + f 2 )X + Y . 24) we deduce that ∇∗X ∇∗Y Y − ∇∗Y ∇∗X Y = 0. 25) with respect to the non–holonomic frame field {X, Y, Z} and obtain [X, Y ] = kf f (−f X + f Y − 2Z). 28) we infer that ∇∗[X,Y ] Y = − 2kf f ∇∗ Y. 28) we get ∇∗Z Y = Q[Z, Y ] = −kf f Q ∂ ∂x1 =− kf f (Y − X).