Abstract
Let $\mathbb {M}^{m_{1}}(c)\times \mathbb {R}^{m_{2}}$ be a Riemannian product of a space form $\mathbb {M}^{m_{1}}(c)$ of constant sectional curvature c and a Euclidean space $\mathbb {R}^{m_{2}}$. Let Mn(n ≥ 2) be an n-dimensional immersed connected submanifold in $\mathbb {M}^{m_{1}}(c)\times \mathbb {R}^{m_{2}}$. We firstly derive the compatible equations for immersion of Mn into $\mathbb {M}^{m_{1}}(c)\times \mathbb {R}^{m_{2}}$. Then, we derive a Simons’ type equation on the squared length of the second fundamental form of Mn. When Mn is compact and minimal in $\mathbb {S}^{m_{1}}(c)\times \mathbb {R}^{m_{2}}$, we prove a series of pinching theorems on the Ricci curvature, the squared length, and the squared maximum norm of the second fundamental form of Mn.
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