Book•
Riemannian Geometry of Contact and Symplectic Manifolds
08 Jan 2002-
TL;DR: In this article, the authors describe a complex geometry model of Symplectic Manifolds with principal S1-bundles and Tangent Sphere Bundles, as well as a negative Xi-sectional Curvature.
Abstract: Preface * 1. Symplectic Manifolds * 2. Principal S1-bundles * 3. Contact Manifolds * 4. Associated Metrics * 5. Integral Submanifolds and Contact Transformations * 6. Sasakian and Cosymplectic Manifolds * 7. Curvature of Contact Metric Manifolds * 8. Submanifolds of Kahler and Sasakian Manifolds * 9. Tangent Bundles and Tangent Sphere Bundles * 10. Curvature Functionals and Spaces of Associated Metrics * 11. Negative Xi-sectional Curvature * 12. Complex Contact Manifolds * 13. Additional Topics in Complex Geometry * 14. 3-Sasakian Manifolds * Bibliography * Subject Index * Author Index
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TL;DR: In this paper, a fundamental exterior differential system of Riemannian geometry was discovered, which is an intrinsic and invariant global system of differential forms of degree n associated to any given oriented Riemmannian manifold M of dimension n+1.
Abstract: We discover a fundamental exterior differential system of Riemannian geometry; indeed, an intrinsic and invariant global system of differential forms of degree n associated to any given oriented Riemannian manifold M of dimension n+1. The framework is that of the tangent sphere bundle of M. We generalise to a Riemannian setting some results from the theory of hypersurfaces in flat Euclidean space. We give new applications and examples of the associated Euler–Lagrange differential systems.
9 citations
Posted Content•
TL;DR: In this paper, the authors developed new techniques for studying concentration of Laplace eigenfunctions as their frequency grows, by decomposing the set of geodesic tubes into those that are non self-looping for time $T$ and those that were.
Abstract: In this article we develop new techniques for studying concentration of Laplace eigenfunctions $\phi_\lambda$ as their frequency, $\lambda$, grows. The method consists of controlling $\phi_\lambda(x)$ by decomposing $\phi_\lambda$ into a superposition of geodesic beams that run through the point $x$. Each beam is localized in phase-space on a tube centered around a geodesic whose radius shrinks slightly slower than $\lambda^{-\frac{1}{2}}$. We control $\phi_\lambda(x)$ by the $L^2$-mass of $\phi_\lambda$ on each geodesic tube and derive a purely dynamical statement through which $\phi_\lambda(x)$ can be studied. In particular, we obtain estimates on $\phi_\lambda(x)$ by decomposing the set of geodesic tubes into those that are non self-looping for time $T$ and those that are. This approach allows for quantitative improvements, in terms of $T$, on the available bounds for $L^\infty$ norms, $L^p$ norms, pointwise Weyl laws, and averages over submanifolds.
9 citations
Journal Article•
TL;DR: In this paper, the curvatures of slant curves in trans-Sasakian manifolds with C-parallel and C-proper mean curvature vector field in the tangent and normal bundles were characterized.
Abstract: We find the characterizations of the curvatures of slant curves in trans-Sasakian manifolds with C-parallel and C-proper mean curvature vector field in the tangent and normal bundles.
9 citations
Cites background from "Riemannian Geometry of Contact and ..."
...Such a manifold is said to be a contact metric manifold if dη = Φ, where Φ(X,Y ) = g(X,φY ) is called the fundamental 2-form of M [4]....
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...and an integral curve of the contact distribution is called a Legendre curve [4]....
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...A normal contact metric manifold is called a Sasakian manifold [4]....
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...Slant curves with contact angle π2 are traditionally called Legendre curves [4]....
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...A (2n+ 1)-dimensional Riemannian manifold M is said to be an almost contact metric manifold [4], if there exist on M a (1, 1) tensor field φ, a vector field ξ, a 1-form η and a Riemannian metric g satisfying φ(2) = −I + η ⊗ ξ, η(ξ) = 1, φξ = 0, η ◦ φ = 0 g(φX,φY ) = g(X,Y )− η(X)η(Y ), η(X) = g(X, ξ), for any vector fields X,Y on M ....
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TL;DR: In this article, it was shown that M2n-1 is a real hypersurface whose geodesics orthogonal to the characteristic vector are mapped to circles of the same curvature 1 in an n-dimensional nonflat complex space form.
Abstract: We show that M2n-1 is a real hypersurface all of whose geodesics orthogonal to the characteristic vector ξ are mapped to circles of the same curvature 1 in an n-dimensional nonflat complex space form $\widetilde{M}_n$(c) (= CPn(c) or CHn(c)) if and only if M is a Sasakian manifold with respect to the almost contact metric structure from the ambient space $\widetilde{M}_n$(c). Moreover, this Sasakian manifold M is a Sasakian space form of constant φ-sectional curvature c + 1 for each c (≠0).
9 citations
TL;DR: In this article, a three-dimensional N(k)-contact metric manifold M admits a Yamabe soliton of type (M,g,V ), and the manifold has a constant scalar curvature and the flow vector field V is Killing.
Abstract: If a three-dimensional N(k)-contact metric manifold M admits a Yamabe soliton of type (M,g,V ), then the manifold has a constant scalar curvature and the flow vector field V is Killing. Furthermore...
9 citations