We establish the following Hadamard–Stoker-type theorem: Let \(f:M^n\rightarrow \mathscr{H} ^{\,\,\, n}\times \mathbb{R}\) be a complete connected hypersurface with positive definite second fundamental form, where \(\mathscr{H} ^{\,\,\, n}\) is a Hadamard manifold. If the height function of f has a critical point, then it is an embedding and M is homeomorphic to \(\mathbb{S}^n\) or \(\mathbb{R}^n.\) Furthermore, f(M) bounds a convex set in \(\mathscr{H} ^{\,\,\, n}\times \mathbb{R}.\) In addition, it is shown that, except for the assumption on convexity, this result is valid for hypersurfaces in \(\mathbb{S}^n\times \mathbb{R}\) as well. We apply these theorems to show that a compact connected hypersurface in \(\mathbb{Q}_{\epsilon}^{n}\times \mathbb{R}\) (\(\epsilon =\pm 1\)) is a rotational sphere, provided it has either constant mean curvature and positive-definite second fundamental form or constant sectional curvature greater than \((\epsilon +1)/2.\) We also prove that, for \(\bar{M}=\mathscr{H} ^{\,\,\, n} \,{\mathrm{or}} \,\, \mathbb{S}^n,\) any connected proper hypersurface \(f:M^n\rightarrow \bar{M}^n \times \mathbb{R}\) with positive semi-definite second fundamental form and height function with no critical points is embedded and isometric to \(\Sigma ^{n-1}\times \mathbb{R},\) where \(\Sigma ^{n-1}\subset \bar{M}^n\) is convex and homeomorphic to \(\mathbb{S}^{n-1}\) (for \(\bar{M}^n=\mathscr{H} ^{\,\,\, n}\) we assume further that f is cylindrically bounded). Analogous theorems for hypersurfaces in warped product spaces \(\mathbb{R}\times _\varrho \mathscr{H} ^{n}\) and \(\mathbb{R}\times _\varrho \mathbb{S}^{n}\) are obtained. In all of these results, the manifold \(M^{n}\) is assumed to have dimension \(n\ge 3.\)

我们建立以下Hadamard-Stoker型定理：令\（f：M ^ n \ rightarrow \ mathscr {H} ^ {\，\，\，n} \ times \ mathbb {R} \）是一个完整的连通超曲面具有肯定的第二基本形式，其中\（\ mathscr {H} ^ {\，\，\，n} \）是Hadamard流形。如果高度函数˚F具有一个临界点，则它是一个嵌入和中号是同胚于\（\ mathbb {S} ^ N \）或\（\ mathbb {R} ^ N \）此外，˚F（中号）限制\（\ mathscr {H} ^ {\，\，\，n} \ times \ mathbb {R}。\）中的凸集。此外，证明了，除了关于凸度的假设之外，该结果对中的超曲面有效\（\ mathbb {S} ^ n \ times \ mathbb {R} \）。我们应用这些定理证明\（\ mathbb {Q} _ {\ epsilon} ^ {n} \ times \ mathbb {R} \）（\（\ epsilon = \ pm 1 \））中的紧连通超曲面是旋转球体，只要它具有恒定的平均曲率和正定的第二基本形式或恒定的截面曲率大于\（（\ epsilon +1）/ 2。\），我们还证明了，对于\（\ bar {M} = \ mathscr {H} ^ {\，\，\，n} \，{\ mathrm {or}} \，\，\ mathbb {S} ^ n，\）任何已连接的适当超曲面\（f：M ^ n \ rightarrow \ bar {M} ^ n \ times \ mathbb {R} \）具有正半定第二基本形式，并且高度函数没有临界点被嵌入，并且等距于\（\ Sigma ^ {n-1} \ times \ mathbb {R}，\）其中\（\ Sigma ^ {n-1} \ subset \ bar {M} ^ n \）是凸且同胚的\（\ mathbb {S} ^ {n-1} \）（对于\（\ bar {M} ^ n = \ mathscr {H} ^ {\，\，\，n} \），我们进一步假设f是圆柱有界的） 。变形产品空间中的超曲面的类似定理\（\ mathbb {R} \ times _ \ varrho \ mathscr {H} ^ {n} \）和\（\ mathbb {R} \ times _ \ varrho \ mathbb {S} ^ {n} \）。在所有这些结果中，流形\（M ^ {n} \）被假定为具有维度\（n \ ge 3. \）。