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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. A note on the Lipschitz classification of Bedford-McMullen carpets
	ZHANG Lu,LUO Jun
	Mathematics 24 February 2022
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	Abstract:The topological structure of fractal sets is a hot topic in the study of fractal geometry and recently there have been a lot of results on the Lipschitz classification of fractal squares. This note mainly considers a class of Bedford-McMullen carpets. By studying the connectedness, the structure of cut-points and so on, this note provides a Lipschitz classification of $20$ different kinds of Bedford-McMullen carpets.
	TO cite this article:ZHANG Lu,LUO Jun. A note on the Lipschitz classification of Bedford-McMullen carpets[OL].[24 February 2022] http://en.paper.edu.cn/en_releasepaper/content/4756449

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	3. Rigidity of polyhedral surfaces with finite boundary components
	Ba Te,Zhou Ze
	Mathematics 08 April 2021
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	Abstract:We prove that the polyhedral surface with finite boundary components is determined up to isometry(or scaling) by some types of discrete curvatures, which generalizes a classical result of Luo Feng. The basic idea is to apply the doubling surgery. In this way, the rigidity of surface with boundary is a corollary of the rigidity of closed surface.
	TO cite this article:Ba Te,Zhou Ze. Rigidity of polyhedral surfaces with finite boundary components[OL].[ 8 April 2021] http://en.paper.edu.cn/en_releasepaper/content/4754475


	4. The sum of Lyapunov exponents on Quadradic differentials
	YU Fei
	Mathematics 26 May 2017
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	Abstract:In this paper we reporve a Kontsevich-Zorich formula for the sum of Lyapunov exponents of Teichm"{u}ller curves on Quadradic differentials. For a Teichm"{u}ller curve in moduli space of abelian differentials. Under some additionalassumptions, we also get an upper bound of individual Lyapunov exponents; in particular we get Lyapunov exponents in hyperelliptic loci and low genusnon-varying strata.
	TO cite this article:YU Fei. The sum of Lyapunov exponents on Quadradic differentials[OL].[26 May 2017] http://en.paper.edu.cn/en_releasepaper/content/4736255


	5. 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


	6. The Mcshane identities for closed geodesics with one self-intersetion on the punctured torus
	LEI Da,ZHANG Ying
	Mathematics 15 May 2017
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	Abstract:This paper gives the Mcshane identities for closed geodesics with one self-intersetion on the punctured torus which is equipped with a complete hyperbolic metric of finite area.
	TO cite this article:LEI Da,ZHANG Ying. The Mcshane identities for closed geodesics with one self-intersetion on the punctured torus[OL].[15 May 2017] http://en.paper.edu.cn/en_releasepaper/content/4734811


	7. Upper bounds of the sum of Lyapunov exponents on Teichmüller curves
	YU Fei
	Mathematics 08 May 2017
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	Abstract:We get anupper bound of the slope of each graded quotient for theHarder-Narasimhan filtration of the Hodge bundle of a Teichmüller curve. As an application, we show that the sum ofLyapunov exponents of a Teichmüller curve does not exceed${(g+1)}/{2}$, with equality reached if and only if the curve liesin the hyperelliptic locus induced from$mathcal{Q}(2k_1,...,2k_n,-1^{2g+2})$ or it is some specialTeichm"{u}ller curve in $Omegamathcal{M}_g(1^{2g-2})$.
	TO cite this article:YU Fei. Upper bounds of the sum of Lyapunov exponents on Teichmüller curves[OL].[ 8 May 2017] http://en.paper.edu.cn/en_releasepaper/content/4733747


	8. Invariant harmonic unit vector fields on oscillator groups
	Tan Ju,Deng Shaoqiang
	Mathematics 06 May 2017
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	Abstract:In this paper, we find all the left-invariant harmonic unit vector fields on the oscillator groups. Besides, we determine the associated harmonic maps from the oscillator group into its unit tangent bundle equipped with the associated Sasaki metric. Moreover, we investigate the stability and instability of harmonic unit vector fields on compact quotients of four dimensional Oscillator group $G_1(1)$.
	TO cite this article:Tan Ju,Deng Shaoqiang. Invariant harmonic unit vector fields on oscillator groups[OL].[ 6 May 2017] http://en.paper.edu.cn/en_releasepaper/content/4733164


	9. Harmonicity of vector fields on a class of Lorentzian solvable Lie groups
	Tan Ju,Deng Shaoqiang
	Mathematics 06 May 2017
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	Abstract:In this paper, we consider a special class of solvable Lie groups such that for any $x, y$ in its Lie algebra, $[x, y]$ is a linear combination of $x$ and $y$. We investigate the harmonicity properties of invariant vector fields of this kind of Lorentzian Lie groups. It is shown that any invariant unit time-like vector field is spatially harmonic. Moreover, we determine all vector fields which are critical points of the energy functional restricted to the space of smooth vector fields.
	TO cite this article:Tan Ju,Deng Shaoqiang. Harmonicity of vector fields on a class of Lorentzian solvable Lie groups[OL].[ 6 May 2017] http://en.paper.edu.cn/en_releasepaper/content/4733161


	10. Volume Growth of Shrinking Gradient Ricci-harmonic soliton
	WU Guoqiang,ZHANG Shi-Jin
	Mathematics 26 April 2017
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	Abstract:In this paper, we study the shrinking gradient Ricci-harmonic soliton. Firstly using Chow-Lu-Yang's argument, we give a necessary and sufficient condition for complete noncompact shrinking gradient Ricci-harmonic solitons with $Sgeq delta$ to have polynomial volume growth with order $n-2delta$. Secondly, we derive a Logarithmic Sobolev inequality, as an application, we prove that any noncompact shrinking gradient Ricci-harmonic soliton must have linear volume growth, generalizing previous result ofMunteanu-Wang.
	TO cite this article:WU Guoqiang,ZHANG Shi-Jin. Volume Growth of Shrinking Gradient Ricci-harmonic soliton[OL].[26 April 2017] http://en.paper.edu.cn/en_releasepaper/content/4728702

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