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1. Isometric immersions of a complete and connected Riemannian manifold | |||
DUAN Jiu-Shun,ZHOU Heng-Yu,ZHOU Heng-Yu | |||
Mathematics 17 January 2023 | |||
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Abstract:In this paper, some basic concepts and theorems of Riemannian manifolds are introduced briefly, and then the concept of isometric immersion is introduced in oeder to introduce the basic cconcept of submanifolds. After introducing Hideki Omori's maximum principle on Riemannian manifolds, a basic theorem of isometric immersion of Riemannian manifolds is proved by using this theorem with modern mathematical language. By replacing $R^n$in the theorem with a more general space and adding additional conditions, the generalized theorem is obtained, and the proof is given in a similar way. | |||
TO cite this article:DUAN Jiu-Shun,ZHOU Heng-Yu,ZHOU Heng-Yu. Isometric immersions of a complete and connected Riemannian manifold[OL].[17 January 2023] http://en.paper.edu.cn/en_releasepaper/content/4758897 |
2. On the Asymptotic upper curvature of hyperbolic products | |||
XIE Gui-Ling,XIAO Ying-Qing | |||
Mathematics 15 May 2017 | |||
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Abstract:M. Bonk and T. Foertsch introduced the notion of asymptotic upper curvature for Gromov hyperbolic spaces and suggested to study the asymptotic upper curvature of hyperbolic products. In this paper, we study these problems and prove that$$K_u(Y_{Delta,o})leqmax{K_u(X_1),K_u(X_2)},$$where $(X_1,o_1),(X_2,o_2)$ are two point Gromov hyperbolic spaces, $Y_{Delta,o}$ is their hyperbolic product and $K_u(X)$ is the asymptotic upper curvature of a hyperbolic space $X$. Moreover, we obtain some extra conditions to sure that $K_u(Y_{Delta,o})$ is no smaller than $K_u(X_2)$. | |||
TO cite this article:XIE Gui-Ling,XIAO Ying-Qing. On the Asymptotic upper curvature of hyperbolic products[OL].[15 May 2017] http://en.paper.edu.cn/en_releasepaper/content/4733628 |
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