Riemannian Geometry of Contact and Symplectic Manifolds
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... ≡ A e r , r = n+1,...,2m+1 are of forms given by (6) and (7). If κ = 1, the (κ,µ)-contact space form reduces to Sasakian space form M˜ (c); thus h = 0 and (2) becomes the equation in Theorem 7.14 of [1]. Therefore, Theorem 3.3 and Theorem 3.4 provide Sasakian versions as Theorem 3.2 of [6] and [2] respectively. Now, we recall Chen’s δ-invariant given by δ M(p) = τ(p) −(inf K)(p) = τ(p) −inf{K(π) | π...
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...bmanifold M of an almost contact metric manifold M˜ with the structure (ϕ,ξ,η,h,i) is called an invariant submanifold if ϕT pM ⊂ T pM. If M˜ is contact also, then ξ ∈ TM, σ(X,ξ) = 0 and M is minimal ([1]). On the other hand, we have the following Proposition 4.1 Every totally umbilical submanifold M of a contact metric manifold such that ξ ∈ TM is minimal and consequently totally geodesic. The proof ...
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...bundle of M and satisfies hX,PYi = −hPX,Y i. Let π ⊂ T pM be a plane section spanned by an orthonormal basis {e1,e2}. Then, α(π) given by α(π) = he1,Pe2i 2 is a real number in the closed unit interval [0,1], which is independent of the choice of the orthonormal basis {e1,e2}. Let ξ ∈ TM and put β(π) = (η(e1))2 +(η(e2))2, γ(π) = η(e1)2 hTe 2,e2 +η(e2)2 hTe 1,e1 −2η(e1)η(e2) hTe 1,e2 . Then, β(π) and γ (π...
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...cteristic examples of non-Sasakian (κ,µ)-contact metric manifolds are the tangent sphere bundles of Riemannian manifolds of constant sectional curvature not equal to one. For more details we refer to [1] and [7]. The sectional curvature K˜ (X,ϕX) of a plane section spanned by a unit vector X orthogonal to ξ is called a ϕ-sectional curvature. If the (κ,µ)-contact metric manifold M˜ has constant ϕ-sect...
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4 citations
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Cites background from "Riemannian Geometry of Contact and ..."
...S 2n+1 with its standard Sasakian structure (see [2], p....
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