Martin Engman
In the special case of S^1 invariant metrics on S^2, we find necessary and sufficient conditions for the existence of isometric embeddings into the canonical R^3, in other words: a Weyl type theorem with converse.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
An upper bound on the first S^1 invariant eigenvalue of the Laplacian for invariant metrics on the 2-sphere is used to find obstructions to the existence of isometric embeddings of such metrics in (R^3,can). As a corollary we prove: If the first four distinct eigenvalues have even multiplicities then the surface of revolution cannot be isometrically embedded in (R^3,can). This leads to a generalization of a classical result in the theory of surfaces.
This note is about a little extension of Nash's embedding theorem in the case of complete manifolds.
Wextend the results obtained recently by G. D'Ambra and A. Loi towards the proof of a conjecture of M.Gromov on isometric immersions via non-free maps.
Notes in the isometric deformations problem in codimension 2
本文通过具体例子研究余维数为2的局部等距刚性问题，指出为保证传递性需扩展真正刚性与诚实刚性概念，并证明必须借助半欧几里得空间中的等距浸入，且该扩展自然引入内积奇异性。
In this paper, we provide families of second order non-linear partial differential equations, describing pseudospherical surfaces (pss equations), with the property of having local isometric immersions in E^3, with principal curvatures depending on finite-order jets of solutions of the differential equation. These equations occupy a particularly special place amongst pss equations, since a series of recent papers [2, 7, 11, 12, 13], on several classes of pss equations, seemed to suggest that only the sine-Gordon equation had the above property. Explicit examples are given, which include the sh…
Let $(M,g)$ be a compact manifold with boundary and $Ric_g\geq (n-1)g$, Hang and Wang proved that $(M,g)$ is isometric to the standard hemisphere if $\partial M$ is convex and isometric to $\mathbb{S}^{n-1}(1)$. We prove some rigidity theorems when $\partial M $ is isometric to a product manifold where one factor is the standard sphere.
This note derives parametrizations for surfaces of revolution that satisfy an affine-linear relation between their respective curvature radii. Alongside, parametrizations for the uniform normal offsets of those surfaces are obtained. Those parametrizations are found explicitly for a countably-infinite many of them, and of those, it is shown which are algebraic. Lastly, for those surfaces which have a constant ratio of principal curvatures, parametrizations with a constant angle between the parameter curves are found.
An extrinsic representation of a Ricci flow on a differentiable n-manifold M is a family of submanifolds S(t), each smoothly embedded in R^{n+k}, evolving as a function of time t such that the metrics induced on the submanifolds S(t) by the ambient Euclidean metric yield the Ricci flow on M. When does such a representation exist? We formulate this question precisely and describe a new, comprehensive way of addressing it for surfaces of revolution in R^3. Of special interest is the Ricci flow on a toroidal surface of revolution, that is, a surface of revolution whose profile curve is an immerse…
In this note, we find a sharp upper bound for the Steklov spectrum on a submanifold of revolution in Euclidean space with one boundary component.
Consider a lattice $Γ$ in a group $G = SL_2(\R), SO(1,n), SU(1,n)$, $SL_2(\Q_p)$. We discuss actions of $Γ$ by affine isometric transformations of Hilbert spaces. We show that for irreducible affine isometric action of $G$ its restriction to $Γ$ is irreducible. We prove the existence of canonical irreducible affine isometric actions of $Γ$ associated to actions of $Γ$ on $\R$- trees. Using such actions we construct irreducible representations of semigroup of probabilistic measures on $Γ$ and construct the series of representations of the groups of diffeomorphisms of Riemann surfaces enumerated…
Quad-surfaces are polyhedral surfaces with quadrilateral faces and the combinatorics of the square grid. A generic quad-surface is rigid. T-hedra is a class of flexible quad-surfaces introduced by Graf and Sauer in 1931. Particular examples of T-hedra are the celebrated Miura fold, discrete surfaces of revolution, and discrete molding surfaces. We provide an explicit parametrization of the isometric deformation of a T-hedron. T-hedra have a smooth analog, T-surfaces. For these we provide a synthetic and an analytic description, both similar to the corresponding descriptions of T-hedra. We also…
We review classical approaches to the problem of isometrically embedding a Riemannian surface into Euclidean 3-space, including coordinate-based approaches exposited by Darboux and Eisenhart, as well as the moving-frames based approaches advocated by Cartan. In particular, the first approach involves reducing the problem to solving a single PDE; settling the question of when this PDE is integrable by the method of Darboux is the subject of this short note. This is surprisingly easy, since it is related to the analogous question for the isometric embedding system arising from the moving frames …
On the geometry of Riemannian isometric embeddings
针对与$\RR^n$微分同胚且允许Bieberbach群$Γ$等距作用的黎曼流形$M$，研究其到欧几里得空间有界子集及$Γ$-等变等距嵌入问题，通过新用已知技巧得到嵌入维数结果，并证明n维光滑流形到$\RR^{2N}$有界子集的嵌入性。
In [4], we proved that every noncompact ancient $κ$-solution to the Ricci flow in dimension $3$ is either locally isometric to a family of shrinking cylinders, or isometric to the Bryant soliton. In the same paper, we announced that the same method implies that compact ancient $κ$-solutions are rotationally symmetric. In this note we provide the details of this argument.
In this note, we show that a nontrivial, compact, degenerate or nondegenerate, gradient Einstein-type manifold of constant scalar curvature is isometric to the standard sphere with a well defined potential function. Moreover, under some geometric assumptions, the noncompact case is also treated. In this case, the main result is that a homogeneous, proper, noncompact, nondegenerate, gradient Einstein-type manifold is an Einstein manifold.
In this note we consider some properties of $GL_n(\mathbb{R})$ with the Semi-Riemannian structure induced by the trace metric $g$. In particular we study geodesics and curvature tensors. Moreover we prove that $GL_n$ has a suitable foliation, whose leaves are isometric to $(SL_n(\mathbb{R}), g)$, while its component of matrices with positive determinant is isometric to the Semi-Riemannian product manifold $SL_n \times \mathbb{R}$.
It was known that if the Gaussian curvature function along each meridian on a surface of revolution $(R^2, dr^2+m(r)^2dθ^2)$ is decreasing, then the cut locus of each point of $θ^{-1}(0)$ is empty or a subarc of the opposite meridian $θ^{-1}(π).$ Such a surface is called a von Mangoldt's surface of revolution. A surface of revolution $(R^2, dr^2+m(r)^2dθ^2)$ is called a generalized von Mangoldt surface of revolution if the cut locus of each point of $θ^{-1}(0)$ is empty or a subarc of the opposite meridian $θ^{-1}(π).$ For example, the surface of revolution $(R^2, dr^2+m_0(r)^2dθ^2),$ where $m…
In this short note we provide an elementary proof of the folklore result in the theory of isometric Lie group actions on Riemannian manifolds asserting that sections of polar actions are totally geodesic.
This short note is a mostly expository article examining negatively curved three-manifolds. We look at some rigidity properties related to isometric embeddings into Minkowski space. We also review the Cross Curvature Flow (XCF) as a tool to study the space of negatively curved metrics on hyperbolic three-manifolds, the largest and least understood class of model geometries in Thurston's Geometrisation. The relationship between integrability and embedability yields interesting insights, and we show that solutions with fixed Einstein volume are precisely the integrable solutions, answering a que…
This note is a contribution to large scale geometry. More precisely, we introduce the intrinsically quasi-isometric sections in metric spaces and we investigate their properties: the Ahlfors-David regularity in large scale; following Cheeger theory, it is possible to define suitable sets in order to obtain convexity and being a vector space over $\mathbb{R}$ or $\mathbb{C}$ for these sections; yet, following Cheeger's idea, we give an equivalence relation for this class of sections. Throughout the paper, we use basic mathematical tools.
The classical Cohn-Vossen theorem states that two isometric compact convex surfaces in $\mathbb{R}^{3}$ are congruent. In this short note, we generalize the classical Cohn-Vossen Theorem to higher dimensional surfaces in space form $N^{n+1}(K)$ for $n\ge 2$.
We consider the Zermelo navigation problem on the ellipsoid of revolution (spheroid) in the presence of a perturbation $W$ determined by a mild velocity vector field, $|W|<1$, with application of Finsler metric of Randers type in the context of the corresponding optimal control represented by a time-efficient ship's heading $φ(t)$ (steering direction). As the example we present the solutions to the problem on an oblate ellipsoid with acting infinitesimal rotation.
A note on the stability of surfaces along null cones under area-preserving variations
研究四维时空零锥类空截面在保面积变化下的稳定性，通过引入相关概念，在主导能量条件下得出稳定截面霍金能量有非负下界等结论，还证明标准闵可夫斯基光锥唯一稳定截面是圆球。
A note on the conformal geometry of the 2-sphere
研究二维球面共形几何，通过特定对称无迹张量导出线性二阶偏微分方程，其解空间有闵可夫斯基结构，存在无穷多相关空间，且二维球面能规范等距嵌入闵可夫斯基空间。
Einstein metrics with torus symmetry on $S^4$ and $\mathbb R^4$
本研究针对S⁴与R⁴上带环面对称性的爱因斯坦度量，通过直接证明得出：有效等距T²作用下，光滑同伦四维球的爱因斯坦度量是圆的，同胚于R⁴的光滑流形上的完备里奇平坦度量经等距、缩放及定向反转后为欧氏度量或单中心Taub–NUT度量。
The first eigenvalue of the Laplacian on a surface can be viewed as a functional on the space of Riemannian metrics of a given area. Critical points of this functional are called extremal metrics. The only known extremal metrics are a round sphere, a standard projective plane, a Clifford torus and an equilateral torus. We construct an extremal metric on a Klein bottle. It is a metric of revolution, admitting a minimal isometric embedding into a 4-sphere by the first eigenfunctions. Also, this Klein bottle is a bipolar surface for the Lawson's {3,1}-torus. We conjecture that an extremal metric …
Let $S$ be a compact, orientable surface of hyperbolic type. Let $(k_+,k_-)$ be a pair of negative numbers and let $(g_+, g_-)$ be a pair of marked metrics over $S$ of constant curvature equal to $k_+$ and $k_-$ respectively. Using a functional introduced by Bonsante, Mondello \& Schlenker, we show that there exists a unique affine deformation $Γ:=(ρ,τ)$ of a Fuchsian group such that $(S,g_+)$ and $(S, g_-)$ embed isometrically as locally strictly convex Cauchy surfaces in the future and past complete components respectively of the quotient by $Γ$ of an open subset $Ω$ of Minkowski space. Such…
The purpose of this note is to scrutinize the proof of Burago and Zalgaller regarding the existence of $PL$ isometric embeddings of $PL$ compact surfaces into $\mathbb{R}^3$. We conclude that their proof does not admit a direct extension to higher dimensions. Moreover, we show that, in general, $PL$ manifolds of dimension $n \geq 3$ admit no nontrivial $PL$ embeddings in $\mathbb{R}^{n+1}$ that are close to conformality. We also extend the result of Burago and Zalgaller to a large class of noncompact $PL$ 2-manifolds. The relation between intrinsic and extrinsic curvatures is also examined, an…
An explicit global and unique isometric embedding into hyperbolic 3-space, H^3, of an axi-symmetric 2-surface with Gaussian curvature bounded below is given. In particular, this allows the embedding into H^3 of surfaces of revolution having negative, but finite, Gaussian curvature at smooth fixed points of the U(1) isometry. As an example, we exhibit the global embedding of the Kerr-Newman event horizon into H^3, for arbitrary values of the angular momentum. For this example, considering a quotient of H^3 by the Picard group, we show that the hyperbolic embedding fits in a fundamental domain o…
Conformal Spherical Splines for the Laplace--Beltrami Operator on Genus-Zero Surfaces: Construction, Algorithm, and Experiments
本文用共形球样条求解零亏格曲面上Laplace--Beltrami算子的Poisson方程和特征值问题，仅需共形因子描述几何，实验验证了最优收敛阶并优于参数曲面有限元。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。