Abstract
It is proved that a Riemannian manifold M isometrically immersed in a Sasakian space form of constant φ‐sectional curvature c < 1, with the structure vector field ξ tangent to M, satisfies Chen′s basic equality if and only if it is a 3‐dimensional minimal invariant submanifold.
Highlights
Let Mbe an m-dimensional almost contact manifold endowed with an almost contact structure (φ, ξ, η), that is, φ be a (1, 1)-tensor field, ξ be a vector field, and η be a 1-form, such that φ2 = −I +η⊗ξ and η(ξ) = 1
An almost contact structure is said to be normal, if in the product manifold M ×R the induced almost complex structure J defined by J(X, λd/dt) = (φX − λξ, η(X)d/dt) is integrable, where X is tangent to M, t is the coordinate of R, and λ is a smooth function on M ×R
Let M be an n-dimensional (n ≥ 3) Riemannian manifold isometrically immersed in a Sasakian space form M (c) of constant φ-sectional curvature c < 1 with the structure vector field ξ tangent to M
Summary
Let Mbe an m-dimensional almost contact manifold endowed with an almost contact structure (φ, ξ, η), that is, φ be a (1, 1)-tensor field, ξ be a vector field, and η be a 1-form, such that φ2 = −I +η⊗ξ and η(ξ) = 1. Φ-sectional curvature c, it is called a Sasakian space form and is denoted by M (c). Let M be an n-dimensional submanifold immersed in an almost contact metric manifold M (φ, ξ, η, g).
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