5
5.0

Jun 28, 2018
06/18

by
H. Azad; Ahmad Y. Al-Dweik; F. M. Mahomed; M. T. Mustafa

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We provide an algorithmic approach to the construction of point transformations for scalar ordinary differential equations that admit three-dimensional symmetry algebras which lead to their respective canonical forms.

Topics: Differential Geometry, Mathematics

Source: http://arxiv.org/abs/1511.02499

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7.0

Jun 29, 2018
06/18

by
Yuxing Deng; Xiaohua Zhu

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In this paper, we prove that any $\kappa$-noncollapsed gradient steady Ricci soliton with nonnegative curvature operator and horizontally $\epsilon$-pinched Ricci curvature must be rotationally symmetric. As an application, we show that any $\kappa$-noncollapsed gradient steady Ricci soliton $(M^n, g,f)$ with nonnegative curvature operator must be rotationally symmetric if it admits a unique equilibrium point and its scalar curvature $R(x)$ satisfies...

Topics: Differential Geometry, Mathematics

Source: http://arxiv.org/abs/1601.02111

4
4.0

Jun 29, 2018
06/18

by
U. Hertrich-Jeromin; A. Honda

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We derive a permutability theorem for the Christoffel, Goursat and Darboux transformations of isothermic surfaces. As a consequence we obtain a simple proof of a relation between Darboux pairs of minimal surfaces in Euclidean space, curved flats in the 2-sphere and flat fronts in hyperbolic space.

Topics: Differential Geometry, Mathematics

Source: http://arxiv.org/abs/1602.06682

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8.0

Jun 29, 2018
06/18

by
J. Seade; K. Shabbir; J. Snoussi

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When f : R power n to R power p, is a surjective real analytic map with isolated critical value, we prove that the (m)-regularity condition (in a sense we define) ensures that f ||f|| is a fibration on small spheres, f induces a fibration on the tubes and both fibrations are equivalent. In particular, we make the statement of [12] more precise in the case of isolated critical point and we extend it to the case of an isolated critical value.

Topics: Differential Geometry, Mathematics

Source: http://arxiv.org/abs/1604.05290

7
7.0

Jun 29, 2018
06/18

by
Andreas Arvanitoyeorgos

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We give an overview of progress on homogeneous Einstein metrics on large classes of homogeneous manifolds, such as generalized flag manifolds and Stiefel manifolds. The main difference between these two classes of homogeneous spaces is that their isotropy representation does not contain/contain equivalent summands. We also discuss a third class of homogeneous spaces that falls into the intersection of such dichotomy, namely the generalized Wallach spaces. We give new invariant Einstein metrics...

Topics: Differential Geometry, Mathematics

Source: http://arxiv.org/abs/1605.05886

6
6.0

Jun 29, 2018
06/18

by
Francisco Urbano

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We classify the homogeneous and isoparametric hypersurfaces of $\mathbb{S}^2\times\mathbb{S}^2$. In the classification, besides the hypersurfaces $\mathbb{S}^1(r)\times\mathbb{S}^2,\,r\in (0,1]$, it appears a family of hypersurfaces with three different constant principal curvatures and zero Gauss-Kronecker curvature. Also we classify the hypersurfaces of $\mathbb{S}^2\times\mathbb{S}^2$ with at most two constant principal curvatures and, under certain conditions, with three constant principal...

Topics: Differential Geometry, Mathematics

Source: http://arxiv.org/abs/1606.07595

5
5.0

Jun 29, 2018
06/18

by
Wolfram Bauer; Kenro Furutani; Chisato Iwasaki

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We construct a co-dimension $3$ completely non-holonomic sub-bundle on the Gromoll-Meyer exotic $7$ sphere based on its realization as a base space of a Sp(2)-principal bundle with the structure group Sp(1). The same method is valid for constructing a co-dimension 3 completely non-holonomic sub-bundle on the standard 7 sphere (or more general on a $4n+3$ dimensional standard sphere). In the latter case such a construction based on the Hopf bundle is well-known. Our method provides an alternated...

Topics: Differential Geometry, Mathematics

Source: http://arxiv.org/abs/1608.02444

4
4.0

Jun 29, 2018
06/18

by
Toshiki Mabuchi

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In this paper, improving a preceding work, we obtain asymptotic polybalanced kernels associated to extremal Kaehler metrics on polarized algebraic manifolds. As a corollary, we have a stronger asymptotic relative Chow-polystability for extremal Kaehler polarized algebraic manifolds. Finally, related to the Yau-Tian-Donaldson Conjecture for extremal Kaehler metrics, we shall discuss the difference between strong relative K-stability and relative K-stability.

Topics: Differential Geometry, Mathematics

Source: http://arxiv.org/abs/1610.09632

6
6.0

Jun 30, 2018
06/18

by
Taiki Yamada

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In this paper, we define the curvature dimension inequalities CD(m, K) on finite directed graphs modifying the case of undirected graphs. As a main result, we evaluate m and K on finite directed graphs.

Topics: Differential Geometry, Mathematics

Source: http://arxiv.org/abs/1701.01510

8
8.0

Jun 30, 2018
06/18

by
Liviu Ornea; Vladimir Slesar

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In this paper we investigate the spectral sequence associated to a Riemannian foliation which arises naturally on a Vaisman manifold. Using the Betti numbers of the underlying manifold we establish a lower bound for the dimension of some terms of this cohomological object. This way we obtain cohomological obstructions for two-dimensional foliations to be induced from a Vaisman structure. We show that if the foliation is quasi-regular the lower bound is realized. In the final part of the paper...

Topics: Differential Geometry, Mathematics

Source: http://arxiv.org/abs/1701.05843

7
7.0

Jun 30, 2018
06/18

by
François Fillastre; Ivan Izmestiev

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By a result of W.~P. Thurston, the moduli space of flat metrics on the sphere with prescribed $n$ cone singularities of positive curvature is a complex hyperbolic orbifold of dimension $n-3$. The Hermitian form comes from the area of the metric. Using geometry of Euclidean polyhedra, we observe that this space has a natural decomposition into real hyperbolic convex polyhedra of dimensions $n-3$ and $\leq \frac{1}{2}(n-1)$. By a result of W.~Veech, the moduli space of flat metrics on a compact...

Topics: Differential Geometry, Mathematics

Source: http://arxiv.org/abs/1702.02114

4
4.0

Jun 30, 2018
06/18

by
Gary R. Jensen; Emilio Musso; Lorenzo Nicolodi

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This note gives sufficient conditions (isothermic or totally nonisothermic) for an immersion of a compact surface to have no Bonnet mate.

Topics: Differential Geometry, Mathematics

Source: http://arxiv.org/abs/1703.09629

4
4.0

Jun 30, 2018
06/18

by
Cristian Ortiz; James Waldron

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In this work we introduce the category of multiplicative sections of an $\la$-groupoid. We prove that this category carries natural strict Lie 2-algebra structures, which are Morita invariant. As applications, we study the algebraic structure underlying multiplicative vector fields on a Lie groupoid and in particular vector fields on differentiable stacks. We also introduce the notion of geometric vector field on the quotient stack of a Lie groupoid, showing that the space of such vector fields...

Topics: Differential Geometry, Mathematics

Source: http://arxiv.org/abs/1703.09791

6
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Jun 30, 2018
06/18

by
Mark Stern

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We study a discrete dynamical system designed to find a 'most holomorphic' connection on a smooth complex vector bundle $E$. We examine the relation between the distance of the chern classes of $E$ from the $(p,p)$ axis of the Hodge diamond and singularity formation. Canonical connections and canonical metrics pulled back from Grassmannians play a major role, and we review their differential geometry. As an exercise in the geometry of canonical connections, we include an expression for the...

Topics: Mathematics, Differential Geometry

Source: http://arxiv.org/abs/1410.8851

10
10.0

Jun 26, 2018
06/18

by
Aldir Brasil; Ezio Costa; Feliciano Vitorio

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This note is devoted to study the implications of nonpositive isotropic curvature and negative Ricci curvature for Einstein $4-$Manifolds.

Topics: Mathematics, Differential Geometry

Source: http://arxiv.org/abs/1502.04401

12
12

Jun 27, 2018
06/18

by
Tongzhu Li; Jie Qing; Changping Wang

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In this paper we show that a Dupin hypersurface with constant M\"{o}bius curvatures is M\"{o}bius equivalent to either an isoparametric hypersurface in the sphere or a cone over an isoparametric hypersurface in a sphere. We also show that a Dupin hypersurface with constant Laguerre curvatures is Laguerre equivalent to a flat Laguerre isoparametric hypersurface. These results solve the major issues related to the conjectures of Cecil et al on the classification of Dupin hypersurfaces.

Topics: Differential Geometry, Mathematics

Source: http://arxiv.org/abs/1503.02914

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14

Jun 28, 2018
06/18

by
Guangyue Huang; Xuerong Qi; Hongjuan Li

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Let $x: M\rightarrow \mathbb{R}^{N}$ be an $n$-dimensional compact self-shrinker in $\mathbb{R}^N$ with smooth boundary $\partial\Omega$. In this paper, we study eigenvalues of the operator $\mathcal{L}_r$ on $M$, where $\mathcal{L}_r$ is defined by $$\mathcal{L}_r=e^{\frac{|x|^2}{2}}{\rm div}(e^{-\frac{|x|^2}{2}}T^r\nabla\cdot)$$ with $T^r$ denoting a positive definite (0,2)-tensor field on $M$. We obtain "universal" inequalities for eigenvalues of the operator $\mathcal{L}_r$. These...

Topics: Differential Geometry, Mathematics

Source: http://arxiv.org/abs/1506.04424

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18

Jun 28, 2018
06/18

by
Sameh Shenawy

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In this note, we introduce the notion of sequential warped product manifolds, to construct a wide variety of warped product manifolds and space-times. First we derive covariant derivative formulas for sequential warped product manifolds. Then we derive many curvature formulas such as curvature tensor, Ricci curvature and scalar curvature formulas. Some important consequences of these formulas are stated. This article also provides characterizations of geodesics and two different types of...

Topics: Differential Geometry, Mathematics

Source: http://arxiv.org/abs/1506.06056

14
14

Jun 28, 2018
06/18

by
Ved Datar; Gábor Székelyhidi

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We show that if a Fano manifold $M$ is K-stable with respect to special degenerations equivariant under a compact group of automorphisms, then $M$ admits a K\"ahler-Einstein metric. This is a strengthening of the solution of the Yau-Tian-Donaldson conjecture for Fano manifolds by Chen-Donaldson-Sun, and can be used to obtain new examples of K\"ahler-Einstein manifolds. We also give analogous results for twisted K\"ahler-Einstein metrics and Kahler-Ricci solitons.

Topics: Differential Geometry, Mathematics

Source: http://arxiv.org/abs/1506.07495

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12

Jun 28, 2018
06/18

by
Benedito Leandro Neto

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In this note we prove that a (anti-)self dual quasi Yamabe soliton with positive sectional curvature is rotationally symmetric. This generalizes a recent result of G. Huang and H. Li in dimension four. Whence, (anti-) self dual gradient Yamabe solitons with positive sectional curvature is rotationally symmetric. We also prove that half conformally flat gradient Yamabe soliton has a special warped product structure provided that the potential function has no critical point.

Topics: Differential Geometry, Mathematics

Source: http://arxiv.org/abs/1507.05974

4
4.0

Jun 28, 2018
06/18

by
Jean-baptiste Casteras; Ilkka Holopainen; Jaime Ripoll

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We study the asymptotic Dirichlet and Plateau problems on Cartan-Hadamard manifolds satisfying the so-called Strict Convexity (abbr. SC) condition. The main part of the paper consists in studying the SC condition on a manifold whose sectional curvatures are bounded from above and below by certain functions depending on the distance to a fixed point. In particular, we are able to verify the SC condition on manifolds whose curvature lower bound can go to -infinity and upper bound to 0...

Topics: Differential Geometry, Mathematics

Source: http://arxiv.org/abs/1507.07311

5
5.0

Jun 28, 2018
06/18

by
Stuart James Hall; Thomas Murphy

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We present a general numerical method for investigating prescribed Ricci curvature problems on toric K\"ahler manifolds. This method is applied to two generalisations of Einstein metrics, namely Ricci solitons and quasi-Einstein metrics. We begin by recovering the Koiso--Cao soliton and the L\"u--Page--Pope quasi-Einstein metrics on $\mathbb{CP}^{2}\sharp\overline{\mathbb{CP}}^{2}$ (in both cases the metrics are known explicitly). We also find numerical approximations to the Wang--Zhu...

Topics: Differential Geometry, Mathematics

Source: http://arxiv.org/abs/1511.03854

8
8.0

Jun 29, 2018
06/18

by
Muhittin Evren Aydin

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A semi-isotropic space is a real affine 3-space endowed with the non-degenerate metric dx^{2}-dy^{2}. The main purpose of this paper is to describe the surfaces of revolution in the semi-isotropic space that satisfy some equations in terms of the position vector and the Laplace operators with respect to the first and the second fundamental forms.

Topics: Differential Geometry, Mathematics

Source: http://arxiv.org/abs/1609.07270

5
5.0

Jun 29, 2018
06/18

by
Julien Roth; Abhitosh Upadhyay

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We study f-biharmonic and bi-f-harmonic submanifolds in both generalized complex and Sasakian space forms. We prove necessary and sufficient condition for f-biharmonicity and bi-f-harmonicity in the general case and many particular cases. Some non-existence results are also obtained.

Topics: Differential Geometry, Mathematics

Source: http://arxiv.org/abs/1609.08599

4
4.0

Jun 29, 2018
06/18

by
Sauvik Mukherjee

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We prove an analogue of Thurston's h-principle for $2$-dimensional foliations on manifolds of dimension bigger or equal to $4$, in the presence of a fiber-wise non-degenerate $2$-form. This helps us understand the flexibility of rank $2$ regular Poisson structures on open manifolds with dimension bigger or equal to $4$ and it also helps us understand the flexibility of Poisson structures (not regular) on closed $4$-manifolds.

Topics: Differential Geometry, Mathematics

Source: http://arxiv.org/abs/1611.09554

7
7.0

Jun 30, 2018
06/18

by
Mohamed Boucetta; Hicham Lebzioui

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A Lorentzian flat Lie group is a Lie group $G$ with a flat left invariant metric $\mu$ with signature $(1,n-1)=(-,+,\ldots,+)$. The Lie algebra $\mathfrak{g}=T_eG$ of $G$ endowed with $\langle\;,\;\rangle=\mu(e)$ is called flat Lorentzian Lie algebra. It is known that the metric of a flat Lorentzian Lie group is geodesically complete if and only if its Lie algebra is unimodular. In this paper, we characterise nonunimodular Lorentzian flat Lie algebras as double extensions (in the sense of...

Topics: Mathematics, Differential Geometry

Source: http://arxiv.org/abs/1401.0950

8
8.0

Jun 30, 2018
06/18

by
Ryosuke Takahashi

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In this paper, we propose a method of studying the modified Kahler-Ricci flow on projective bundles and give the explicit equation from the view point of symplectic geometry.

Topics: Mathematics, Differential Geometry

Source: http://arxiv.org/abs/1403.0985

7
7.0

Jun 30, 2018
06/18

by
E. Loubeau; C. Oniciuc

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CMC surfaces in spheres are investigated under the extra condition of biharmonicity. From the work of Miyata, especially in the flat case, we give a complete description of such immersions and show that for any $h\in (0,1)$ there exist CMC proper-biharmonic planes and cylinders in $\sn^5$ with $|H|=h$, while a necessary and sufficient condition on $h$ is found for the existence of CMC proper-biharmonic tori in $\sn^5$.

Topics: Mathematics, Differential Geometry

Source: http://arxiv.org/abs/1403.1703

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10.0

Jun 30, 2018
06/18

by
Zhiqi Chen; Yifang Kang; Ke Liang

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In this paper, we classify three-locally-symmetric spaces for a connected, compact and simple Lie group. Furthermore, we give the classification of invariant Einstein metrics on these spaces.

Topics: Mathematics, Differential Geometry

Source: http://arxiv.org/abs/1411.2694

4
4.0

Jun 30, 2018
06/18

by
V. Berestovskii; I. Zubareva

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The authors compute distances between arbitrary elements of Lie groups SU(2) and SO(3) for special left-invariant sub-Riemannian metrics $\rho$ and $d$. To compute distances for the second metric, we essentially use the fact that canonical two-sheeted covering epimorphism $\Omega$ of the Lie group SU(2) onto the Lie group SO(3) is submetry and local isometry with respect to metrics $\rho$ and $d$. Proofs are based on previously known formulas for geodesics with origin at the unit, F. Klein's...

Topics: Mathematics, Differential Geometry

Source: http://arxiv.org/abs/1411.4718

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19

Jun 30, 2018
06/18

by
Barnabe Pessoa Lima; Paulo Alexandre Araujo Sousa; Juscelino Pereira Silva; Newton Luis Santos

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In this paper, we give a complete description of all translation hypersurfaces with constant r-curvature Sr, in the Euclidean space.

Topics: Mathematics, Differential Geometry

Source: http://arxiv.org/abs/1402.2362

5
5.0

Jun 30, 2018
06/18

by
Sergio Augusto Romaña Ibarra

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In a paper from 1954, Marstrand proved that if $K\subset \mathbb{R}^2$ with Hausdorff dimension greater than 1, then its one-dimensional projection has positive Lebesgue measure for almost-all directions. In this article, we show that if $M$ is a simply connected surface with non-positive curvature, then Marstrand's theorem is still valid.

Topics: Mathematics, Differential Geometry

Source: http://arxiv.org/abs/1402.5133

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4.0

Jun 30, 2018
06/18

by
Naoyuki Koike

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In this paper, we prove that, if a full irreducible infinite dimensional anti-Kaehler isoparametric submanifold of codimension greater than one has $J$-diagonalizable shape operators, then it is homogeneous.

Topics: Mathematics, Differential Geometry

Source: http://arxiv.org/abs/1404.0822

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6.0

Jun 30, 2018
06/18

by
Hristo Manev

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The space of the structure (0,3)-tensors of the covariant derivatives of the structure endomorphism and the metric on almost contact B-metric manifolds is considered. A known decomposition of this space in orthogonal and invariant subspaces with respect to the action of the structure group is used. We determine the corresponding components of the structure tensor and consider the case of the lowest dimension 3 of the studied manifolds. Some examples are commented.

Topics: Mathematics, Differential Geometry

Source: http://arxiv.org/abs/1405.3088

3
3.0

Jun 30, 2018
06/18

by
Sugata Mondal

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Let $S$ be a noncompact, finite area hyperbolic surface of type $(g, n)$. Let $\Delta_S$ denote the Laplace operator on $S$. As $S$ varies over the {\it moduli space} ${\mathcal{M}_{g, n}}$ of finite area hyperbolic surfaces of type $(g, n)$, we study, adapting methods of Lizhen Ji \cite{Ji} and Scott Wolpert \cite{Wo}, the behavior of {\it small cuspidal eigenpairs} of $\Delta_S$. In Theorem 2 we describe limiting behavior of these eigenpairs on surfaces ${S_m} \in {\mathcal{M}_{g, n}}$ when...

Topics: Mathematics, Differential Geometry

Source: http://arxiv.org/abs/1406.1076

4
4.0

Jun 30, 2018
06/18

by
Giovanni Catino; Paolo Mastrolia; Dario D. Monticelli

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In this short note we prove that, in dimension three, flat metrics are the only complete metrics with non-negative scalar curvature which are critical for the $\sigma_{2}$-curvature functional.

Topics: Mathematics, Differential Geometry

Source: http://arxiv.org/abs/1406.0990

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5.0

Jun 30, 2018
06/18

by
Ming Xu; Shaoqiang Deng; Libing Huang; Zhiguang Hu

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In this paper, we use the technique of Finslerian submersion to deduce a flag curvature formula for homogeneous Finsler spaces. Based on this formula, we give a complete classification of even-dimensional smooth coset spaces $G/H$ admitting $G$-invariant Finsler metrics with positive flag curvature. It turns out that the classification list coincides with that of the even dimensional homogeneous Riemannian manifolds with positive sectional curvature obtained by N.R. Wallach. We also find out...

Topics: Mathematics, Differential Geometry

Source: http://arxiv.org/abs/1407.3582

3
3.0

Jun 30, 2018
06/18

by
Haggai Nuchi

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Heinz Hopf's famous fibrations of the 2n+1-sphere by great circles, the 4n+3-sphere by great 3-spheres, and the 15-sphere by great 7-spheres have a number of interesting properties. Besides providing the first examples of homotopically nontrivial maps from one sphere to another sphere of lower dimension, they all share two striking features: (1) Their fibers are parallel, in the sense that any two fibers are a constant distance apart, and (2) The fibrations are highly symmetric. For example,...

Topics: Mathematics, Differential Geometry

Source: http://arxiv.org/abs/1407.4549

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4.0

Jun 30, 2018
06/18

by
Miriam Telichevesky

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Given an unbounded domain $\Omega$ of a Hadamard manifold $M$, it makes sense to consider the problem of finding minimal graphs with prescribed continuous data on its cone-topology-boundary, i.e., on its ordinary boundary together with its asymptotic boundary. In this article it is proved that under the hypothesis that the sectional curvature of $M$ is $\le -1$ this Dirichlet problem is solvable if $\Omega$ satisfies certain convexity condition at infinity and if $\partial \Omega$ is mean...

Topics: Mathematics, Differential Geometry

Source: http://arxiv.org/abs/1409.5155

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9.0

Jun 30, 2018
06/18

by
Jia-Yong Wu; Peng Wu; William Wylie

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We prove that a four-dimensional gradient shrinking Ricci soliton with $\delta W^{\pm}=0$ is either Einstein, or a finite quotient of $S^3\times\mathbb{R}$, $S^2\times\mathbb{R}^2$ or $\mathbb{R}^4$. We also prove that a four-dimensional cscK gradient Ricci soliton is either K\"ahler-Einstein, or a finite quotient of $M\times\mathbb{C}$, where $M$ is a Riemann surface. The main arguments are curvature decompositions, the Weitzenb\"ock formula for half Weyl curvature, and the maximum...

Topics: Mathematics, Differential Geometry

Source: http://arxiv.org/abs/1410.7303

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3.0

Jun 30, 2018
06/18

by
David Blázquez-Sanz; Juan Sebastián Díaz Arboleda

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We discuss the general properties of the theory of joint invariants of a smooth Lie group action in a manifold. Many of the known results about differential invariants, including Lie's finiteness theorem, have simpler versions in the context of joint invariants. We explore the relation between joint and differential invariants, and we expose a general method that allow to compute differential invariants from joint invariants.

Topics: Mathematics, Differential Geometry

Source: http://arxiv.org/abs/1410.7878

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9.0

Jun 30, 2018
06/18

by
P. Gilkey; C. Y. Kim; J. H. Park

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We use the solution space of a pair of ODEs of at least second order to construct a smooth surface in Euclidean space. We describe when this surface is a proper embedding which is geodesically complete with finite total Gauss curvature. If the associated roots of the ODEs are real and distinct, we give a universal upper bound for the total Gauss curvature of the surface which depends only on the orders of the ODEs and we show that the total Gauss curvature of the surface vanishes if the ODEs...

Topics: Mathematics, Differential Geometry

Source: http://arxiv.org/abs/1411.0487

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8.0

Jun 30, 2018
06/18

by
Pilar Herreros; Mario Ponce; J. J. P Veerman

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For distinct points $p$ and $q$ in a two-dimensional Riemannian manifold, one defines their mediatrix $L_{pq}$ as the set of equidistant points to $p$ and $q$. It is known that mediatrices have a cell decomposition consisting of a finite number of branch points connected by Lipschitz curves. This paper establishes additional geometric regularity properties of mediatrices. We show that mediatrices have the radial linearizability property, which implies that at each point they have a...

Topics: Mathematics, Differential Geometry

Source: http://arxiv.org/abs/1411.1803

3
3.0

Jun 30, 2018
06/18

by
Lorenzo Foscolo

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We show that for generic choices of parameters the moduli spaces of periodic monopoles (with singularities), i.e. monopoles on $\mathbb{R}^{2} \times \mathbb{S}^{1}$ possibly singular at a finite collection of points, are either empty or smooth hyperk\"ahler manifolds. Furthermore, we prove an index theorem and therefore compute the dimension of the moduli spaces.

Topics: Mathematics, Differential Geometry

Source: http://arxiv.org/abs/1411.6946

3
3.0

Jun 30, 2018
06/18

by
Y. Keshavarzi

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Our aim in this paper is to investigate some geometrical properties of Berger Spheres i.e. homogeneous Ricci solitons and harmonicity properties of invariant vector fields. We determine all vector fields which are critical points for the energy functional restricted to vector fields. We also see that do not exist any vector fields defining harmonic map, and the energy of critical points is explicitly calculated.

Topics: Mathematics, Differential Geometry

Source: http://arxiv.org/abs/1412.6336

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10.0

Jun 27, 2018
06/18

by
Abraham Henrique Muñoz Flores; Stefano Nardulli

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We study the problem of existence of isoperimetric regions for large volumes, in $C^0$-locally asymptotically Euclidean Riemannian manifolds with a finite number of $C^0$-asymptotically Schwarzschild ends. Then we give a geometric characterization of these isoperimetric regions, extending previous results contained in [EM13b], [EM13a], and [BE13]. Moreover strengthening a little bit the speed of convergence to the Schwarzschild metric we obtain existence of isoperimetric regions for all volumes...

Topics: Differential Geometry, Mathematics

Source: http://arxiv.org/abs/1503.02361

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7.0

Jun 27, 2018
06/18

by
Xin Nie

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We show that the volume entropy of the Hilbert metric on a closed convex projective surface tends to zero as the corresponding Pick differential tends to infinity. The proof is based on the theorem, due to Benoist and Hulin, that the Hilbert metric and Blaschke metric are comparable.

Topics: Differential Geometry, Mathematics

Source: http://arxiv.org/abs/1503.04420

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8.0

Jun 27, 2018
06/18

by
Sorin V. Sabau

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We show that the co-rays to a ray in a complete non-compact Finsler manifold contain geodesic segments to upper level sets of Busemann functions. Moreover, we characterise the co-point set to a ray as the cut locus of such level sets. The structure theorem of the co-point set on a surface, namely that is a local tree, and other properties follow immediately from the known results about the cut locus. We point out that some of our findings, in special the relation of co-point set to the upper...

Topics: Differential Geometry, Mathematics

Source: http://arxiv.org/abs/1504.03921

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10.0

Jun 27, 2018
06/18

by
Xiaoli Han; Jiayu Li; Jun Sun

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In this paper we derive the Euler-Lagrange equation of the functional $L_\beta=\int_\Sigma\frac{1}{\cos^\beta\alpha}d\mu, ~~\beta\neq -1$ in the class of symplectic surfaces. It is $\cos^3\alpha {\bf{H}}=\beta(J(J\nabla\cos\alpha)^\top)^\bot$, which is an elliptic equation when $\beta\geq 0$. We call such a surface a $\beta$-symplectic critical surface. We first study the properties for each fixed $\beta$-symplectic critical surface and then prove that the set of $\beta$ where there is a stable...

Topics: Differential Geometry, Mathematics

Source: http://arxiv.org/abs/1504.04138

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3.0

Jun 30, 2018
06/18

by
Andreas Hermann; Emmanuel Humbert

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Let $(M,g)$ be a closed Riemannian manifold of dimension $n \geq 3$ and let $f\in C^{\infty}(M)$, such that the operator $P_f:= \Delta_g+f$ is positive. If $g$ is flat near some point $p$ and $f$ vanishes around $p$, we can define the mass of $P_f$ as the constant term in the expansion of the Green function of $P_f$ at $p$. In this paper, we establish many results on the mass of such operators. In particular, if $f:= \frac{n-2}{4(n-1)} \scal_g$, i.e. if $P_f$ is the Yamabe operator, we show the...

Topics: Mathematics, Differential Geometry

Source: http://arxiv.org/abs/1401.1614