13 research outputs found
Rank and k-nullity of contact manifolds
No full textWe prove that the dimension of the 1-nullity distribution N(1) on a closed Sasakian manifold M of rankl is at least equal to 2l−1 provided that M has an isolated closed characteristic. The result is then used to provide some examples of k-contact manifolds which are not Sasakian. On a closed, 2n+1-dimensional Sasakian manifold of positive bisectional curvature, we show that either the dimension of N(1) is less than or equal to n+1 or N(1) is the entire tangent bundle TM. In the latter case, the Sasakian manifold Mis isometric to a quotient of the Euclidean sphere under a finite group of isometries. We also point out some interactions between k-nullity, Weinstein conjecture, and minimal unit vector fields
Generalized Eta-Einstein and -structures
No full textGeneralized structures occur in dimension 3 only. In this
dimension 3, only K-contact structures can occur as generalized Eta-Einstein.
On closed manifolds, Eta-Einstein, K-contact structures which are not
D-homothetic to Einstein structures are almost regular. We also construct
examples of compact, generalized Jacobi -structures
© Hindawi Publishing Corp. RANK AND k-NULLITY OF CONTACT MANIFOLDS
No full textWe prove that the dimension of the 1-nullity distributionN(1) on a closed Sasakian manifold M of rank l is at least equal to 2l−1 provided thatM has an isolated closed characteristic. The result is then used to provide some examples ofK-contact manifolds which are not Sasakian. On a closed, 2n+ 1-dimensional Sasakian manifold of positive bisectional curvature, we show that either the dimension of N(1) is less than or equal to n+1 or N(1) is the entire tangent bundle TM. In the latter case, the Sasakian manifoldM is isometric to a quotient of the Euclidean sphere under a finite group of isometries. We also point out some interactions between k-nullity, Weinstein conjecture, and minimal unit vector fields. 2000 Mathematics Subject Classification: 53D35. 1. Introduction. Contac
are satisfied for any vector fields X and Y on 1M. Such a metric g is called a contact metric. On a compact (2n + 1)-dimensional contact metric manifold (M, g, (Y), n> 1, the contact metric g cannot be flat. In dimension 3 however, flat contact metrics do exist, as the following example shows (111): In local coordinates ~1, x2, x3, the standard contact form cr on T3 is given by cr = ~(COS x3dxl + sin 2ad22). Its characteristic vector field is < = 2(cos x3 & + sinza&) and its flat contact metric is gij = a6i.j. The contact form Q is invariant under all translations with vector of the type (a, b, k27r). This form cr is also invariant under any screw motion (Ro, tp) where Re is a rotation of angle 0 in the (zi, x2) plane and t, is a translation of vector p = (0,0,2n-8). IfR 0 is a rotation which preserves a lattice containing (a, b, 27r), then cr induces a contact form on the quotient T3 / < Re, t,>, where < Rs, t,> is the cyclic group generated by (Re
Fibrations and contact structures
No full textWe prove that a closed 3-dimensional manifold is a torus bundle over the circle if and only if it carries a closed nonsingular 1-form which is linearly deformable into contact forms
