A DDVV Type Inequality and a Pinching Theorem for Compact Minimal Submanifolds in a Generalized Cylinder $$\mathbb {S}^{n_1}(c)\times \mathbb {R}^{n_2}$$

Abstract

In this paper, we prove a new DDVV type inequality for submanifolds immersed in a Riemannian product $$\mathbb {M}^{n_1}(c)\times \mathbb {R}^{n_2} (c\ne 0)$$ of a real space form $$\mathbb {M}^{n_1}(c)$$ of curvature c and a Euclidean space $$\mathbb {R}^{n_2}$$ . In addition, we obtain a pinching theorem for compact minimal submanifolds immersed in a generalized cylinder $$\mathbb {S}^{n_1}(c)\times \mathbb {R}^{n_2}$$ of a sphere $$\mathbb {S}^{n_1}(c)$$ with curvature $$c>0$$ and a Euclidean space $$\mathbb {R}^{n_2}$$ , which extends Lu’s and Chen–Cui’s pinching theorems.