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**

**Sample text**

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.