By David E. Blair
This monograph bargains with the Riemannian geometry of either symplectic and call manifolds, with specific emphasis at the latter. The textual content is punctiliously offered. themes spread systematically from bankruptcy 1, which examines the final thought of symplectic manifolds. central circle bundles (Chapter 2) are then mentioned as a prelude to the Boothby--Wang fibration of a compact typical touch manifold in bankruptcy three, which offers with the final idea of touch manifolds. bankruptcy four makes a speciality of the overall environment of Riemannian metrics linked to either symplectic and make contact with buildings, and bankruptcy five is dedicated to fundamental submanifolds of the touch subbundle. issues taken care of within the next chapters comprise Sasakian manifolds, the real 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 houses 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 path in Riemannian geometry.
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Additional resources for Riemannian Geometry of Contact and Symplectic Manifolds
To see this first note that the tangent space to the orbit of g under the action of S at g is the set of tensor fields of the form £xg where X is a symplectic vector field. 5) Xi = Jikek for some closed 1-form 0. 7. 1 we considered R2n+1 with its usual contact structure dz 1 yidxi and saw that the contact subbundle D is spanned by -mL +y2 a y;, i = 1 ... n. For normalization convenience, we take as the standard contact structure on R2n+1 the 1-form 77 = 2 (dz - E2 1 yidxi). The characteristic vector field is then = 2-2- and the Riemannian metric ii ((dxi)2+(dyi)2) i=1 gives a contact metric structure on R2n+1.
2 for details). 5 T*M X R Let M be an n-dimensional manifold and T*M its cotangent bundle. As in the previous example we can define a 1-form Q by the local expression Y 1 pidgi. Let Men+1 = T*M x R, t the coordinate on R and y : Men+1 T*M the projection to the first factor. 6 T3 We have mentioned that Martinet proved that every compact orientable 3manifold carries a contact structure. Here we will give explicitly a contact structure on the 3-dimensional torus V. First consider R3 with the contact form 77 = sin ydx + cos ydz; i7 A dry = -dx A dy A dz.
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.
Riemannian Geometry of Contact and Symplectic Manifolds by David E. Blair