Download Riemannian Geometry of Contact and Symplectic Manifolds by David E. Blair PDF
By David E. Blair
This monograph offers with the Riemannian geometry of either symplectic and phone manifolds, with specific emphasis at the latter. The textual content is thoroughly awarded. themes spread systematically from bankruptcy 1, which examines the overall conception of symplectic manifolds. significant circle bundles (Chapter 2) are then mentioned as a prelude to the Boothby--Wang fibration of a compact usual touch manifold in bankruptcy three, which bargains with the overall conception of touch manifolds. bankruptcy four specializes in the final surroundings of Riemannian metrics linked to either symplectic and call constructions, and bankruptcy five is dedicated to critical submanifolds of the touch subbundle. subject matters handled within the next chapters contain Sasakian manifolds, the $64000 learn of the curvature of touch metric manifolds, submanifold concept in either the K¿hler and Sasakian settings, tangent sphere bundles, curvature functionals, advanced touch manifolds and three Sasakian manifolds. The publication serves either as a normal reference for mathematicians to the fundamental homes of symplectic and touch manifolds and as a very good source for graduate scholars and researchers within the Riemannian geometric area. The prerequisite for this article is a uncomplicated direction in Riemannian geometry.
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Extra resources for Riemannian Geometry of Contact and Symplectic Manifolds
Example text
Indeed this is the case and we now show that the set A of all associated metrics is infinite dimensional by exhibiting a path of metrics in A determined by a C°° function with compact support. Such paths of associated metrics will be useful to us in the study of critical points of curvature functionals on A. We give the construction in the symplectic case and then remark on the similarity with the contact case. Let f be a C°° function with compact support contained in a neighborhood U of (M2n, Q) and {X1, ...
An almost complex structure is a tensor field J of type (1,1) such that J2 = -I. , g(X, JY) _ -g(JX, Y), S2(X, Y) = g(X, JY) defines a 2-form called the fundamental 2form of the almost Hermitian structure (M, J, g). If M is a complex manifold and J the corresponding almost complex structure, we say that (M, J, g) is a Hermitian manifold. If df = 0, the structure is almost Kdhler. For geometers working strictly over the complex domain, a Hermitian metric is a Hermitian quadratic form and hence complex-valued; it takes its non-zero values as appropriate when one argument is holomorphic and the other anti-holomorphic.
Let {Lfa} be an open cover of M2n+1 by coordinate charts. Recall the local contact forms rla above and their transition 77a = Taprjp. Consider the M2n+1 given as follows. M2,+1 is the union of 2-fold covering 7r : M2n+l the sets {U,,, X Z2} with the following equivalence relation, where Z2 is the set of integers ±1. Let ea denote +1 or -1. Two elements (pa, Ea) E U", X Z2 and (pp, Ep) E Up X Z2 are equivalent if pa = pp and ea = sgn(Tap(pp))Ep. Now define local contact forms on ua x ±1 by rl(a Eo) = Ea7r*rla.