Euclidean volume growth for complete Riemannian manifolds

平面特殊欧几里得群中的度量线

Metric Lines in the Special Euclidean group on the plane

本文研究平面特殊欧几里得群SE(2)中的子黎曼流形，探讨周期子黎曼测地线的必要条件及度量线的性质，采用哈密顿-雅可比理论提供新证明。

Biharmonic submanifolds in a Riemannian manifold

In this paper, we solve affirmatively B.-Y. Chen's conjecture for hypersurfaces in the Euclidean space, under a generic condition. More precisely, every biharmonic hypersurface of the Euclidean space must be minimal if their principal curvatures are simple, and the associated frame field is irreducible.

Classification of conformal solitons in pseudo-Euclidean spaces

In this paper, we completely classify conformal solitons on pseudo-Riemannian hypersurfaces in pseudo-Euclidean spaces arisen from the position vector field. In particular, the classification of Yamabe solitons on pseudo-Riemannian hypersurfaces in pseudo-Euclidean spaces arisen from the position vector field can be obtained.

Riemannian surfaces with torsion as homogenization limits of locally-Euclidean surfaces with dislocation-type singularities

We reconcile between two classical models of edge-dislocations in solids. The first model, dating from the early 1900s models isolated edge-dislocations as line singularities in locally-Euclidean manifolds. The second model, dating from the 1950s, models continuously-distributed edge-dislocations as smooth manifolds endowed with non-symmetric affine connections (equivalently, endowed with torsion fields). In both models, the solid is modeled as a Weitzenböck manifold. We prove, using a weak notion of convergence [KM15], that the second model can be obtained rigorously as a homogenization limit…

Isometric immersions of Riemannian manifolds in $k$-codimensional Euclidean space

We use a new method to give conditions for the existence of a local isometric immersion of a Riemannian $n$-manifold $M$ in $\mathbb{R}^{n+k}$, for a given $n$ and $k$. These equate to the (local) existence of a $k$-tuple of scalar fields on the manifold, satisfying a certain non-linear equation involving the Riemannian curvature tensor of $M$. Setting $k=1$, we proceed to recover the fundamental theorem of hypersurfaces. In the case of manifolds of positive sectional curvature and $n\geq 3$, we reduce the solvability of the Gauss and Codazzi equations to the cancelation of a set of obstructio…

k-harmonic maps into a Riemannian manifold with constant sectional curvature

J. Eells and L. Lemaire introduced k-harmonic maps, and Wang Shaobo showed the first variational formula. When, k=2, it is called biharmonic maps (2-harmonic maps). There have been extensive studies in the area. In this paper, we consider the relationship between biharmonic maps and k-harmonic maps, and show non-existence theorem of 3-harmonic maps. We also give the definition of k-harmonic submanifolds of Euclidean spaces, and study k-harmonic curve in Euclidean spaces. Futhermore, we give a conjecture for k-harmonic submanifolds of Euclidean spaces.

Ricci Bounds for Euclidean and Spherical Cones

We prove generalized lower Ricci bounds for Euclidean and spherical cones over compact Riemannian manifolds. These cones are regarded as complete metric measure spaces. We show that the Euclidean cone over an n-dimensional Riemannian manifold whose Ricci curvature is bounded from below by n-1 satisfies the curvature-dimension condition CD(0,n+1) and that the spherical cone over the same manifold fulfills the curvature-dimension condition CD(n,n+1).

Growth conditions for conformal transformations preserving Riemannian completeness

For a complete Riemannian metric, a pointwise conformal transformation may lead to a complete or incomplete transformed Riemannian metric, depending on the behavior of the conformal factor. We establish conditions on the growth of the conformal factor towards the infinity of the Riemannian metric, such that the conformally transformed Riemannian metric remains complete.

Algebraic nature of singular Riemannian foliations in spheres

We prove that singular Riemannian foliations in Euclidean spheres can be defined by polynomial equations.

A geometric perspective on the Piola identity in Riemannian settings

The Piola identity $\operatorname{div} \operatorname{cof} \nabla f=0$ is a central result in the mathematical theory of elasticity. We prove a generalized version of the Piola identity for mappings between Riemannian manifolds, using two approaches, based on different interpretations of the cofactor of a linear map: one follows the lines of the classical Euclidean derivation and the other is based on a variational interpretation via Null-Lagrangians. In both cases, we first review the Euclidean case before proceeding to the general Riemannian setting.

Sub-Riemannian curvature and a Gauss-Bonnet theorem in the Heisenberg group

We use a Riemannnian approximation scheme to define a notion of $\textit{sub-Riemannian Gaussian curvature}$ for a Euclidean $C^{2}$-smooth surface in the Heisenberg group $\mathbb{H}$ away from characteristic points, and a notion of $\textit{sub-Riemannian signed geodesic curvature}$ for Euclidean $C^{2}$-smooth curves on surfaces. These results are then used to prove a Heisenberg version of the Gauss-Bonnet theorem. An application to Steiner's formula for the Carnot-Carathéodory distance in $\mathbb{H}$ is provided.

Growth of the eigensolutions of Laplacians on Riemannian manifolds II: positivity of the initial energy

In this paper, energy function is used to investigate the eigen-solutions of $-Δu+ Vu=λu$ on the Riemannian manifolds. We give a new way to prove the positivity of the initial energy of energy function, which leads to a simple way to obtain the growth of eigen-solutions.

Isothermic submanifolds of Euclidean space

We study the problem posed by F. Burstall of developing a theory of isothermic Euclidean submanifolds of dimension greater than or equal to three. As a natural extension of the definition in the surface case, we call a Euclidean submanifold {\it isothermic} if it is locally the image of a conformal immersion of a Riemannian product of Riemannian manifolds whose second fundamental form is adapted to the product net of the manifold. Our main result is a complete classification of all such conformal immersions of Riemannian products of dimension greater than or equal to three. We derive several c…

Classification of Ricci solitons on Euclidean hypersurfaces

A Ricci soliton $(M,g,v,λ)$ on a Riemannian manifold $(M,g)$ is said to have concurrent potential field if its potential field $v$ is a concurrent vector field. Ricci solitons arisen from concurrent vector fields on Riemannian manifolds were studied recently in \cite{CD2}. The most important concurrent vector field is the position vector field on Euclidean submanifolds. In this paper we completely classify Ricci solitons on Euclidean hypersurfaces arisen from the position vector field of the hypersurfaces.

具有欧几里得体积增长的黎曼流形上的精确估计与不等式

Sharp estimates and inequalities on Riemannian manifolds with Euclidean volume growth

该研究针对具有欧几里得体积增长、里奇曲率非负的完备非紧黎曼流形，推导热核与格林函数的精确估计，并借此得到流形上的精确Moser-Trudinger不等式。

Polynomial growth of subharmonic functions in a strongly symmetric Riemannian manifold

In this article we have studied some properties of subharmonic functions in a strongly symmetric Riemannian manifold with a pole. As a generalization of polynomial growth of a function we have introduced the notion of polynomial growth of some degree of a function with respect to a real function and proved that any non-negative twice differentiable subharmonic functions in an $n$-dimensional manifold always admit polynomial growth of degree $1$ with respect to a non-negative real valued subharmonic function on real line. We have also given a lower bound of the integration of a convex function …

从黎曼流形到非退化欧氏锥的映射

Maps from Riemannian manifolds into non-degenerate Euclidean cones

本文研究连通非紧黎曼流形到非退化欧氏锥的光滑映射，基于能量、张力与度量参数给出锥宽下界，并推广了调和映射、极小浸入与 Kähler 子流形的经典结论及等距浸入的排除性结果。

Riemannian Rigidity of the Parallel Postulate in Total Curvature

We prove that the Euclidean plane is the only Riemannian plane with total curvature and free of conjugate points that satisfies Playfair's version of the parallel postulate.

A Compactness Theorem for Rotationally Symmetric Riemannian Manifolds with Positive Scalar Curvature

Gromov and Sormani conjectured that sequences of compact Riemannian manifolds with nonnegative scalar curvature and area of minimal surfaces bounded below should have subsequences which converge in the intrinsic flat sense to limit spaces which have nonnegative generalized scalar curvature and Euclidean tangent cones almost everywhere. In this paper we prove this conjecture for sequences of rotationally symmetric warped product manifolds. We show that the limit spaces have $H^1$ warping function that has nonnegative scalar curvature in a weak sense, and have Euclidean tangent cones almost ever…

海森堡群中的闵可夫斯基问题

The Minkowski problem in Heisenberg groups

本文首次在海森堡群的水平分布中提出并解决了次黎曼版本的闵可夫斯基型问题，通过变分方法给出了肯定回答。

Stability and the index of biharmonic hypersurfaces in a Riemannian manifold

In this paper, we give an explicit second variation formula for a biharmonic hypersurface in a Riamannian manifold similar to that of a minimal hypersurface. We then use the second variation formula to compute the stability index of the known biharmonic hypersurfaces in a Euclidean sphere, and to prove the non-existence of unstable proper biharmonic hypersurface in a Euclidean space or a hyperbolic space, which adds another special case to support Chen's conjecture on biharmonic submanifolds.

About the Injectivity Radius and the Ricci Tensor of a Complete Riemannian Manifold

In this paper we obtain a simple upper bound for the infimum of the Ricci curvatures of a complete Riemannian manifold with nonzero injectivity radius i(M) depending only on of the i(M). In case of rigidity the Riemannian manifold must be an Euclidean sphere(Euclidean space) conform the injectivity radius be finite(infinite). Furthermore with the additional assumption that the second derivative of the Ricci tensor is null we prove that the same upper bound for the infimum of the Ricci curvatures holds for the supremum of the Ricci curvatures and $M^n$ has, in fact, parallel Ricci tensor.

Metric Foliations on the Euclidean Space

We complete a minor gap in Gromoll and Walschap classification of metric fibrations from the Euclidean space, thus completing the classification of Riemannian foliations on Euclidean spaces.

Classification of codimension two deformations of rank two Riemannian manifolds

The purpose of this work is to close the local deformation problem of rank two Euclidean submanifolds in codimension two by describing their moduli space of deformations. In the process, we provide an explicit simple representation of these submanifolds, a result of independent interest by its applications. We also determine which deformations are genuine and honest, allowing us to find the first known examples of honestly locally deformable rank two submanifolds in codimension two. In addition, we study which of these submanifolds admit isometric immersions as Euclidean hypersurfaces, a prope…

A family of warped product semi-Riemannian Einstein metrics

We study warped products semi-Riemannian Einstein manifolds. We consider the case in that the base is conformal to an n-dimensional pseudo Euclidean space and invariant under the action of an translation group. We provide all such solutions in the case Ricci flat when the base is conformal to an n-dimensional pseudo-Euclidean space, invariant under the action of an translation group and the fiber F is Ricci flat. In particular, we obtain explicit solutions, in the case vacuum, for the Einstein field equation.

Chen's conjecture and epsilon-superbiharmonic submanifolds of Riemannian manifolds

B.-Y. Chen famously conjectured that every submanifold of Euclidean space with harmonic mean curvature vector is minimal. In this note we establish a much more general statement for a large class of submanifolds satisfying a growth condition at infinity. We discuss in particular two popular competing natural interpretations of the conjecture when the Euclidean background space is replaced by an arbitrary Riemannian manifold. Introducing the notion of epsilon-superbiharmonic submanifolds, which contains each of the previous notions as special cases, we prove that epsilon-superbiharmonic submani…

A Riemannian approach to the membrane limit in non-Euclidean elasticity

Non-Euclidean, or incompatible elasticity is an elastic theory for pre-stressed materials, which is based on a modeling of the elastic body as a Riemannian manifold. In this paper we derive a dimensionally-reduced model of the so-called membrane limit of a thin incompatible body. By generalizing classical dimension reduction techniques to the Riemannian setting, we are able to prove a general theorem that applies to an elastic body of arbitrary dimension, arbitrary slender dimension, and arbitrary metric. The limiting model implies the minimization of an integral functional defined over immers…

Minimal helix submanifolds and Minimal Riemannian foliations

We investigate minimal helix submanifolds of any dimension and codimension immersed in Euclidean space. Our main result proves that a ruled minimal helix submanifold is a cylinder. As an application we classify complex helix submanifolds of $\mathbb{C}^n$: They are extrinsic products with a complex line as a factor. The key tool is Corollary 1.3 which allows us to classify Riemannian foliations of open subsets of the Euclidean space with minimal leaves. Finally, we consider the case of a helix hypersurface with constant mean curvature and prove that it is either a cylinder or an open part of a…

Warped product pointwise bi-slant submanifolds in metallic Riemannian manifolds

In this paper, we study some properties of warped product pointwise bi-slant submanifolds in locally metallic Riemannian manifolds and we construct some examples in Euclidean spaces.

Geodesic Convexity Types in Riemannian Manifolds

The usual notion of set-convexity, valid in the classical Euclidean context, metamorphoses into several distinct convexity types in the more general Riemannian setting. By studying this phenomenon in reverse, we characterize complete manifolds for which certain convexity types are assumed a priori to coincide.

Some existence results to the Dirichlet problem for the minimal hypersurface equation on non mean convex domains of a Riemannian manifold

We prove the existence of minimal hypersurfaces for the Dirichlet that extends a similar result of Jenkins and Serrin in Euclidean Space to Riemannian ambient manifolds

等距浸入于欧几里得空间中黎曼流形上散度形式算子特征值的不等式

Inequalities for eigenvalues of operators in divergence form on Riemannian manifolds isometrically immersed in Euclidean space

本文研究了等距浸入于欧几里得空间的黎曼流形上散度形式二阶椭圆算子的特征值问题，通过建立Yang型第一不等式推导出普遍不等式，并证明了支持特殊函数和张量的流形的相关不等式。

Pseudo-Riemannian geodesics and billiards

Many classical facts in Riemannian geometry have their pseudo-Riemannian analogs. For instance, the spaces of space-like and time-like geodesics on a pseudo-Riemannian manifold have natural symplectic structures (just like in the Riemannian case), while the space of light-like geodesics has a natural contact structure. We discuss the geometry of these structures in detail, as well as introduce and study pseudo-Euclidean billiards. In particular, we prove pseudo-Euclidean analogs of the Jacobi-Chasles theorems and show the integrability of the billiard in the ellipsoid and the geodesic flow on …

The classification of flat Riemannian metrics on the plane

We classify all smooth flat Riemannian metrics on the two-dimensional plane. In the complete case, it is well-known that these metrics are isometric to the Euclidean metric. In the incomplete case, there is an abundance of naturally-arising, non-isometric metrics that are relevant and useful. Remarkably, the study and classification of all flat Riemannian metrics on the plane -- as a subject -- is new to the literature. Much of our research focuses on conformal metrics of the form $e^{2φ}g_0$, where $φ: \mathbb{R}^2 \rightarrow \mathbb{R)$ is a harmonic function and $g_0$ is the standard Eucli…

On the fundamental group of complete manifolds with almost Euclidean volume growth

It is proved that the fundamental group of a complete Riemannian manifold with nonnegative Ricci curvature and certain volume growth conditions is trivial or finite.

等距浸入欧几里得空间的黎曼流形上一类散度型椭圆微分算子的特征值估计

Eigenvalue estimates for a class of elliptic differential operators in divergence form on Riemannian manifolds isometrically immersed in Euclidean space

该研究针对等距浸入欧氏空间的完备黎曼流形有界域上一类更广的散度型椭圆微分算子推导特征值估计，将其应用于高斯收缩孤立子并找到估计表现类拉普拉斯情形的区域，还解答了Pólya广义猜想。

Some classifications of \infty-Harmonic maps between Riemannian manifolds

$\infty$-Harmonic maps are a generalization of $\infty$-harmonic functions. They can be viewed as the limiting cases of p-harmonic maps as p goes to infinity. In this paper, we give complete classifications of linear and quadratic $\infty$-harmonic maps from and into a sphere, quadratic $\infty$-harmonic maps between Euclidean spaces. We describe all linear and quadratic $\infty$-harmonic maps between Nil and Euclidean spaces, between Sol and Euclidean spaces. We also study holomorphic $\infty$-harmonic maps between complex Euclidean spaces.

On stable compact minimal submanifolds of Riemannian product manifolds

In this paper, we prove a classification theorem for the stable compact minimal submanifolds of the Riemannian product of an $m_1$-dimensional ($m_1\geq3$) hypersurface $M_1$ in the Euclidean space and any Riemannian manifold $M_2$, when the sectional curvature $K_{M_1}$ of $M_1$ satisfies $\frac{1}{\sqrt{m_1-1}}\leq K_{M_1}\leq 1.$ This gives a generalization to the results of F. Torralbo and F. Urbano [9], where they obtained a classification theorem for the stable minimal submanifolds of the Riemannian product of a sphere and any Riemannian manifold. In particular, when the ambient space is…

Intrinsic Riemannian metrics on spaces of curves: theory and computation

This chapter reviews some past and recent developments in shape comparison and analysis of curves based on the computation of intrinsic Riemannian metrics on the space of curves modulo shape-preserving transformations. We summarize the general construction and theoretical properties of quotient elastic metrics for Euclidean as well as non-Euclidean curves before considering the special case of the square root velocity metric for which the expression of the resulting distance simplifies through a particular transformation. We then examine different numerical approaches that have been proposed t…

欧氏空间扰动的连续正质量定理

A continuous Positive Mass Theorem for perturbations of Euclidean space

本文证明了对于与欧氏空间足够接近且具有弱非负标量曲率的$C^0$-渐近平坦度量，其$C^0$类比ADM质量非负，推广了正质量定理。

Chen's conjecture on biharmonic submanifolds in Riemannian manifolds

We study biharmonic hypersurfaces and biharmonic submanifolds in a Riemannian manifold. One of interesting problems in this direction is Chen's conjecture which says that any biharmonic submanifold in a Euclidean space is minimal. From the invariant equation for biharmonic submanifolds, we derive a fundamental identity involving the mean curvature vector field, and using this, we prove Chen's conjecture on biharmonic submanifolds in a Euclidean space. More generally, it is proved that any biharmonic submanifold in a space form of nonpositively sectional curvature is minimal. Furthermore we pro…

Harmonic vector fields on pseudo-Riemannian manifolds

The theory of harmonic vector fields on Riemannian manifolds is generalised to pseudo-Riemannian manifolds. Harmonic conformal gradient fields on pseudo-Euclidean hyperquadrics are classified up to congruence, as are harmonic Killing fields on pseudo-Riemannian quadrics. A para-Kaehler twisted anti-isometry is used to correlate harmonic vector fields on the quadrics of neutral signature.

The geometry of the Sasaki metric on the sphere bundle of Euclidean Atiyah vector bundles

Let $(M,\langle,\rangle_{TM})$ be a Riemannian manifold. It is well-known that the Sasaki metric on $TM$ is very rigid but it has nice properties when restricted to $T^{(r)}M=\{u\in TM,|u|=r \}$. In this paper, we consider a general situation where we replace $TM$ by a vector bundle $E\longrightarrow M$ endowed with a Euclidean product $\langle,\rangle_E$ and a connection $\nabla^E$ which preserves $\langle,\rangle_E$. We define the Sasaki metric on $E$ and we consider its restriction $h$ to $E^{(r)}=\{a\in E,\langle a,a\rangle_E=r^2 \}$. We study the Riemannian geometry of $(E^{(r)},h)$ gener…

Growth of the eigensolutions of Laplacians on Riemannian manifolds I: construction of energy function

In this paper, we consider the eigen-solutions of $-Δu+ Vu=λu$, where $Δ$ is the Laplacian on a non-compact complete Riemannian manifold. We develop Kato's methods on manifold and establish the growth of the eigen-solutions as $r$ goes to infinity based on the asymptotical behaviors of $Δr$ and $V(x)$, where $r=r(x)$ is the distance function on the manifold. As applications, we prove several criteria of absence of eigenvalues of Laplacian, including a new proof of the absence of eigenvalues embedded into the essential spectra of free Laplacian if the radial curvature of the manifold satisfies …

On local isometric embeddings of three-dimensional Lie groups

Due to Janet-Cartan's theorem, any analytic Riemannian manifolds can be locally isometrically embedded into a sufficiently high dimensional Euclidean space. However, for an individual Riemannian manifold (M,g), it is in general hard to determine the least dimensional Euclidean space into which (M,g) can be locally isometrically embedded, even in the case where (M,g) is homogeneous. In this paper, when the space (M,g) is locally isometric to a three-dimensional Lie group equipped with a left-invariant Riemannian metric, we classify all such spaces that can be locally isometrically embedded into…

Some variational properties of the weighted σ_{r}-curvature for submanifolds in Riemannian manifolds

The objet of this paper is the study of the variations of a functional whose integrant is the r-th weighted curvature on the hypersurface of a closed Riemannian manifold. Some applications to hypersurfaces of the Euclidean space and the unit round sphère are given.

Trudinger-Moser inequalities on complete noncompact Riemannian manifolds

Though Trudinger-Moser inequalities on compact Riemannian manifolds or Euclidean space are well understood, we know little about them on complete noncompact Riemannian manifolds. In this paper, we established respectively necessary condition and sufficient condition under which Trudinger-Moser inequalities hold on complete noncompact Riemannian manifolds. For applications of such inequalities, we obtained existence results for some quasilinear equations with nonlinearity of exponential growth.

Characterizations of umbilic points of isometric immersions in Riemannian and Lorentzian manifolds

Several characterizations of umbilic points of submanifolds in arbitrary Riemannian and Lorentzian manifolds are given. As a consequence, we obtain new characterizations of spheres in the Euclidean space and of hyperbolic spaces in the Lorentz-Minkowski space. We also prove the Lorentzian version of a classical result by Cartan.

Induced structures on submanifolds in almost product Riemannian manifolds

We give some fundamental properties of the induced structures on submanifolds immersed in almost product or locally product Riemannian manifolds. We study the induced structure by the composition of two isometric immersions on submanifolds in an almost product Riemannian manifold. We give an effective construction for some induced structures on submanifolds of codimension 1 or 2 in Euclidean space.

Polyharmonic maps into the Euclidean space

We study polyharmonic (k-harmonic) maps between Riemannian manifolds with finite j-energies (j=1, cdots, 2k-2). We show if the domain is complete and the target is the Euclidean space, then such a map is harmonic.

带欧几里得约束的黎曼quasi-Newton算法

A Riemannian quasi-Newton algorithm for optimization with Euclidean bounds

本文提出一种结合有限记忆quasi-Newton更新和黎曼适应的广义Cauchy点策略的黎曼有限记忆BFGS方法，用于处理欧几里得约束的优化问题，展示了在协方差矩阵估计和神经影像等领域中的高效性。

Kropina metrics and Zermelo navigation on Riemannian manifolds

The present paper studies globally defined Kropina metrics as solutions of the Zermelo's navigation problem. Moreover, we characterize the Kropina metrics of constant flag curvature showing that up to local isometry, there are only two model spaces of them: the Euclidean space and the odd-dimensional spheres.

Exploration of Balanced Metrics on Symmetric Positive Definite Matrices

Symmetric Positive Definite (SPD) matrices have been used in many fields of medical data analysis. Many Riemannian metrics have been defined on this manifold but the choice of the Riemannian structure lacks a set of principles that could lead one to choose properly the metric. This drives us to introduce the principle of balanced metrics that relate the affine-invariant metric with the Euclidean and inverse-Euclidean metric, or the Bogoliubov-Kubo-Mori metric with the Euclidean and log-Euclidean metrics. We introduce two new families of balanced metrics, the mixed-power-Euclidean and the mixed…

Monge-Ampère type equations with Neumann boundary conditions on Riemannian manifolds

In this paper, we consider the global regularity for Monge-Ampère type equations with the Neumann boundary conditions on Riemannian manifolds. It is known that the classical solvability of the Neumann boundary value problem is obtained under some necessary assumptions. Our main result extends the main theorem from the case of Euclidean space $R^n$ in [11] to Riemannian manifolds.