Download Riemannian Geometry of Contact and Symplectic Manifolds by David E. Blair PDF

By David E. Blair

This moment variation, divided into fourteen chapters, provides a complete remedy of touch and symplectic manifolds from the Riemannian standpoint. The monograph examines the elemental principles intimately and offers many illustrative examples for the reader.

Riemannian Geometry of touch and Symplectic Manifolds, moment Edition presents new fabric in such a lot chapters, yet a specific emphasis is still on touch manifolds. New vital issues comprise a fancy geodesic stream and the accompanying geometry of the projectivized holomorphic tangent package and a fancy model of the specified instructions mentioned in bankruptcy eleven for the genuine case. either one of those themes utilize Étienne Ghys's appealing thought of a holomorphic Anosov flow.

Researchers, mathematicians, and graduate scholars in touch and symplectic manifold conception and in Riemannian geometry will reap the benefits of this paintings. A simple direction in Riemannian geometry is a prerequisite.

Reviews from the 1st Edition:

"The booklet . . . can be utilized both as an advent to the topic or as a reference for college students and researchers . . . [it] supplies a transparent and entire account of the most rules . . . and reports an enormous quantity of similar matters reminiscent of quintessential sub-manifolds, symplectic constitution of tangent bundles, curvature of touch metric manifolds and curvature functionals on areas of linked metrics." —Mathematical Reviews

"…this is a delightful and precious ebook and all geometers will revenue [from] studying it. they could use it for complex classes, for thesis themes in addition to for references. rookies will locate in it an enticing [table of] contents and necessary rules for pursuing their studies." —Memoriile Sectiilor Stiintifice

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Extra resources for Riemannian Geometry of Contact and Symplectic Manifolds

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Note in particular that an almost complex manifold is even-dimensional. Again given (M, J), choose g Hermitian. Let {Uα } be an open cover ¯i , X ¯i∗ } on Uα and Uβ respectively. With rewith J-bases {Xi , Xi∗ }, {X spect to these bases J is given by 0 −I I 0 . 1 Almost complex and almost contact structures where A, B, C, D are n × n matrices and 0 −I I 0 A B C D = A B C D A B C D (X) = (JX) = 0 −I I 0 43 ∈ O(2n). Now A B C D (JX) (X), and I0 −I commute. Therefore D = A and C = −B and hence 0 U (n).

Is vertical. Thus X Given a connection H on (P, G, M ) define a g-valued 1-form φ on P ¯ by φ(t) = X ∈ g where X(p) is the vertical part of t. φ is called the connection form of H. , Bishop and Crittenden [1964, pp. 76–77], Kobayashi and Nomizu [1963–69, Volume I, p. 64]). , φ is equivariant. 2 If φ is a g-valued C ∞ equivariant 1-form such that ¯ φ(X(p)) = X, then there exists a unique connection H whose 1-form is φ. Given a g-valued p-form σ and a g-valued q-form ω, their bracket is defined by [σ, ω](X1 , .

R ∂θ ∂z ∂ are Legendre Note that in both examples the integral curves of ∂r curves and in the second example that the curve r = π, z = const. is also a Legendre curve. 2 Examples 33 disk Δ = {z = 0, r ≤ π} ⊂ R3 along the boundary. Now consider the topological disk Δ = {z = r 2 , r ≤ π}. D is tangent to Δ only at the origin. On Δ \ {(0, 0, 0)} define a line field by the intersection of the tangent plane to the paraboloid Δ and D at each point. These fields can be expressed by the vector fields r ∂ ∂ ∂ +2 + 2 r2 ∂r ∂θ ∂z in the first case and r sin r ∂ ∂ ∂ − 2 r cos r + 2 r 2 sin r ∂r ∂θ ∂z in the second.

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