Differentiable sphere theorems for submanifolds of positive k-th ricci curvature

Abstract

We investigate the differentiable pinching problem for compact immersed submanifolds of positive k-th Ricci curvature, and prove that if M n is simply connected and the k-th Ricci curvature of M n is bounded below by a quantity involving the mean curvature of M n and the curvature of the ambient manifold, then M n is diffeomorphic to the standard sphere S n . For the case where the ambient manifold is a space form with nonnegative constant curvature, we prove a differentiable sphere theorem without the assumption that the submanifold M n is simply connected. Motivated by a geometric rigidity theorem due to S. T. Yau and U. Simon, we prove a topological rigidity theorem for submanifolds in a space form.

Homology Groups in Warped Product Submanifolds in Hyperbolic Spaces

“…e geometric structure and topological properties of submanifolds in different spaces have been studied on a large scale during the past few years [4][5][6][7][8][9][10][11][12][13][14][15][16][17]. Many results showed that there is a closed relationship between stable currents which are nonexistent and the vanished homology groups of submanifolds in a different class of the ambient manifold obtained by imposing conditions on the second fundamental form (1).…”

Geometric Inequalities of Warped Product Submanifolds and Their Applications

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Sectional and intermediate Ricci curvature lower bounds via optimal transport

“…It is convenient to also set σ Proof. If we set y x (t) = log J x (t) = log detB x (t), from (23) in Proposition 4.4 we know that y ′′ x (t) + 1 p y ′ x (t) 2 + Ric p (T γx(t) Σ t ,γ x (t)) − U ⊥ (t) 2 ≤ 0, ∀t ∈ (0, 1), ∀x ∈ Σ 1/2 \ N.…”