Nonzero Constant Curvature Research Articles

Some classifications of 2-dimensional self-shrinkers

It is well-known that self-shrinkers play an important role in the study of mean curvature flows. In this paper, we develop new techniques to study the rigidity of self-shrinkers. We prove that any self-shrinker X:M→R3 with nonzero constant Gauss curvature is the round sphere S2(2). Moreover, we prove that any flat self-shrinker X:M→R3 is a plane R2, a cylinder S1(1)×R, a generalized cylinder Γ×R, where Γ is an Abresch-Langer curve. At last, we show that any self-shrinker X:M→R3 with constant mean curvature is a plane R2, a cylinder S1(1)×R or the round sphere S2(2).

On concurrent vector fields on Riemannian manifolds

<abstract><p>It is shown that the presence of a non-zero concurrent vector field on a Riemannian manifold poses an obstruction to its topology as well as certain aspects of its geometry. It is shown that on a compact Riemannian manifold, there does not exist a non-zero concurrent vector field. Also, it is shown that a Riemannian manifold of non-zero constant scalar curvature does not admit a non-zero concurrent vector field. It is also shown that a non-zero concurrent vector field annihilates de-Rham Laplace operator. Finally, we find a characterization of a Euclidean space using a non-zero concurrent vector field on a complete and connected Riemannian manifold.</p></abstract>

Covariant derivatives of eigenfunctions along parallel tensors over space forms and a conjecture motivated by the vertex algebraic structure

We study the covariant derivatives of an eigenfunction for the Laplace–Beltrami operator on a complete, connected Riemannian manifold with nonzero constant sectional curvature. We show that along every parallel tensor, the covariant derivative is a scalar multiple of the eigenfunction. We also show that the scalar is a polynomial depending on the eigenvalue and prove some properties. A conjecture motivated by the study of vertex algebraic structure on space forms is also announced, suggesting the existence of interesting structures in these polynomials that awaits further exploration.

Infinitely long isotropic Kirchhoff rods with helical centerlines cannot be stable.

It has long been known that every configuration of a planar elastic rod with clamped ends satisfies the property that if its centerline has constant nonzero curvature, then it is in stable equilibrium regardless of its length. In this paper, we show that for a certain class of nonplanar elastic rods, no configuration satisfies this property. In particular, using results from optimal control theory, we show that every configuration of an inextensible, unshearable, isotropic, and uniform Kirchhoff rod with clamped ends that has a helical centerline with constant nonzero curvature becomes unstable at a finite length. We also derive coordinates for computing this critical length that are independent of the rod's bending and torsional stiffness. Finally, we derive a scaling relationship between the length at which a helical rod becomes unstable and the rod's curvature, torsion, and twist. In a companion paper, these results are used to compute the set of all stable rods with helical centerlines.

On Complete Conformally Flat Submanifolds with Nullity in Euclidean Space

In this note, we investigate conformally flat submanifolds of Euclidean space with positive index of relative nullity. Let $$M^n$$ be a complete conformally flat manifold and let $$f:M^n\rightarrow \mathord {\mathbb {R}}^m$$ be an isometric immersion. We prove the following results: (1) If the index of relative nullity is at least two, then $$M^n$$ is flat and f is a cylinder over a flat submanifold. (2) If the scalar curvature of $$M^n$$ is non-negative and the index of relative nullity is positive, then f is a cylinder over a submanifold with constant non-negative sectional curvature. (3) If the scalar curvature of $$M^n$$ is non-zero and the index of relative nullity is constant and equal to one, then f is a cylinder over a $$(n-1)$$ -dimensional submanifold with non-zero constant sectional curvature.

Геометрия риманова пространства второго приближения

Для риманова пространства ненулевой постоянной кривизны Vn построено приближение второго порядка - пространство V2n. Доказано, что V2n является субпроективным пространством В. Ф. Кагана. В явном виде получено выражение компонент вектора Киллинга пространства V2n.

Semi-discrete linear Weingarten surfaces with Weierstrass-type representations and their singularities

We establish what semi-discrete linear Weingarten surfaces with Weierstrass-type representations in $3$-dimensional Riemannian and Lorentzian spaceforms are, confirming their required properties regarding curvatures and parallel surfaces, and then classify them. We then define and analyze their singularities. In particular, we discuss singularities of (1) semi-discrete surfaces with non-zero constant Gaussian curvature, (2) parallel surfaces of semi-discrete minimal and maximal surfaces, and (3) semi-discrete constant mean curvature $1$ surfaces in de Sitter $3$-space. We include comparisons with different previously known definitions of such singularities.

Existence of conformal metrics with constant scalar curvature and constant boundary mean curvature on compact manifolds

We study the problem of deforming a Riemannian metric to a conformal one with nonzero constant scalar curvature and nonzero constant boundary mean curvature on a compact manifold of dimension [Formula: see text]. We prove the existence of such conformal metrics in the cases of [Formula: see text] or the manifold is spin and some other remaining ones left by Escobar. Furthermore, in the positive Yamabe constant case, by normalizing the scalar curvature to be [Formula: see text], there exists a sequence of conformal metrics such that their constant boundary mean curvatures go to [Formula: see text].

Classification of the orthogonal separable webs for the Hamilton-Jacobi and Laplace-Beltrami equations on 3-dimensional hyperbolic and de Sitter spaces

We review the theory of orthogonal separation of variables on pseudo-Riemannian manifolds of constant non-zero curvature via concircular tensors and warped products. We then apply this theory simultaneously to both the three-dimensional hyperbolic and de Sitter spaces, obtaining an invariant classification of the thirty-four orthogonal separable webs on each space, modulo action of the respective isometry groups. The inequivalent coordinate charts adapted to each web are also determined and listed. The results obtained for hyperbolic 3-space agree with those in the literature, while the results for de Sitter 3-space appear to be new.

On T-Slant, N-Slant and B-Slant helices in galilean space 𝔾3

In this paper, we define T-slant, N-slant and B-slant helices in Galilean space 𝔾3. In particular, we obtain the explicit parameter equations of the T-slant helices and prove that an admissible curve is a T-slant helix with a non-isotropic axis if and only if it has a non-zero constant conical curvature. We also prove that there are no N-slant, B-slant and Darboux helices in the same space.

Convexity, Rigidity, and Reduction of Codimension of Isometric Immersions into Space Forms

We consider isometric immersions of complete connected Riemannian manifolds into space forms of nonzero constant curvature. We prove that if such an immersion is compact and has semi-definite second fundamental form, then it is an embedding with codimension one, its image bounds a convex set, and it is rigid. This result generalizes previous ones by do Carmo and Lima, as well as by do Carmo and Warner. It also settles affirmatively a conjecture by do Carmo and Warner. We establish a similar result for complete isometric immersions satisfying a stronger condition on the second fundamental form. We extend to the context of isometric immersions in space forms a classical theorem for Euclidean hypersurfaces due to Hadamard. In this same context, we prove an existence theorem for hypersurfaces with prescribed boundary and vanishing Gauss-Kronecker curvature. Finally, we show that isometric immersions into space forms which are regular outside the set of totally geodesic points admit a reduction of codimension to one.

Central Configurations of the Curved N-Body Problem

We consider the $N$-body problem of celestial mechanics in spaces of nonzero constant curvature. Using the concept of locked inertia tensor, we compute the moment of inertia for systems moving on spheres and hyperbolic spheres and show that we can recover the classical definition in the Euclidean case. After proving some criteria for the existence of relative equilibria, we find a natural way to define the concept of central configuration in curved spaces using the moment of inertia, and show that our definition is formally similar to the one that governs the classical problem. The existence criteria we develop for central configurations help us provide several examples and prove that, for any given point masses on spheres and hyperbolic spheres, central configurations always exist. We end our paper with results concerning the number of central configurations that lie on the same geodesic, thus extending the celebrated theorem of Moulton to hyperbolic spheres and pointing out that it has no straightforward generalization to spheres, where the count gets complicated even in the case $N=2$.

Reduction and relative equilibria for the two-body problem on spaces of constant curvature

We consider the two-body problem on surfaces of constant non-zero curvature and classify the relative equilibria and their stability. On the hyperbolic plane, for each q>0 we show there are two relative equilibria where the masses are separated by a distance q. One of these is geometrically of elliptic type and the other of hyperbolic type. The hyperbolic ones are always unstable, while the elliptic ones are stable when sufficiently close, but unstable when far apart. On the sphere of positive curvature, if the masses are different, there is a unique relative equilibrium (RE) for every angular separation except {\pi}/2. When the angle is acute, the RE is elliptic, and when it is obtuse the RE can be either elliptic or linearly unstable. We show using a KAM argument that the acute ones are almost always nonlinearly stable. If the masses are equal there are two families of relative equilibria: one where the masses are at equal angles with the axis of rotation (`isosceles RE') and the other when the two masses subtend a right angle at the centre of the sphere. The isosceles RE are elliptic if the angle subtended by the particles is acute and is unstable if it is obtuse. At {\pi}/2, the two families meet and a pitchfork bifurcation takes place. Right-angled RE are elliptic away from the bifurcation point. In each of the two geometric settings, we use a global reduction to eliminate the group of symmetries and analyse the resulting reduced equations which live on a 5-dimensional phase space and possess one Casimir function.

On Stretch curvature of Finsler manifolds

In this paper, Finsler metrics with relatively non-negative (resp. non-positive), isotropic and constant stretch curvature are studied. In particular, it is showed that every compact Finsler manifold with relatively non-positive (resp. non-negative) stretch curvature is a Landsberg metric. Also, it is proved that every (α,β)-metric of non-zero constant flag curvature and non-zero relatively isotropic stretch curvature on a manifold of dimension n>2 has a constant characteristic scalar along the geodesics. Two dimensional Finsler manifolds of relatively stretch curvature are studied, too../files/site1/files/0Abstract4.pdf

Wavelet analysis on some smooth surface with nonzero constant Gaussian curvature

According to the theory of wavelet analysis on [Formula: see text], the wavelet analysis on a smooth surface [Formula: see text] with nonzero constant Gaussian curvature will be discussed systematically in this paper. First, a general area-preserving projection from a smooth surface to the plane will be presented by the Gaussian projection and the area-preserving projection on the sphere. Then the continuous wavelet transform and its inverse transform on a smooth surface [Formula: see text] with nonzero constant Gaussian curvature will be discussed by a general area-preserving projection, relative dilation operator and translation operator. Further, according to the multi-resolution analysis on a smooth surface, the discrete wavelet transform and relative properties will be investigated systematically, including the two-scale equations of the wavelet function, orthogonality and so on. Finally, two numerical examples will be given.

On the Geometry of Metric Measure Spaces with Variable Curvature Bounds

Motivated by a classical comparison result of J. C. F. Sturm, we introduce a curvature-dimension condition CD(k, N) for general metric measure spaces, variable lower curvature bound \(k\) and upper dimension bound \(N\ge 1\). In the case of non-zero constant lower curvature, our approach coincides with the celebrated condition that was proposed by Sturm (Acta Math 196(1):133–177, 2006). We prove several geometric properties as sharp Bishop–Gromov volume growth comparison or a sharp generalized Bonnet–Myers theorem (Schneider’s Theorem). In addition, the curvature-dimension condition is stable with respect to measured Gromov–Hausdorff convergence, and it is stable with respect to tensorization of finitely many metric measure spaces provided a non-branching condition is assumed. We also briefly describe possible extensions for variable dimension bounds.

Erratum to: On the geometry of spaces of oriented geodesics

1. Those entries of Table 1 in the original paper listing the numbers of geometric structures on the space of geodesics L (S p,q) of the 3-dimensional space S p,q of non-zero constant curvature with p+q = 3 are incorrect. In this note we give the correct values and arguments. In original paper we failed to note that the (so(V )+so(V ′))-module V ⊗V ′, where V, V ′ are 2-dimensional pseudo-Euclidian spaces, has not only the invariant (para)complex structures J = JV ⊗ 1, J ′ = 1 ⊗ JV ′ , but that also their product J ′′ = J J ′ = JV ⊗ JV ′ is an invariant (para)complex structure. We next analyze the impact of this observation in detail and correct Table 1 accordingly. The authors would like to thank Henri Anciaux for drawing their attention to this oversight. 2. Let E = E p+1,q be a pseudo-Euclidean 4-dimensional vector space of signature (p + 1, q) = (4, 0), (3, 1), (2, 2) or (1, 3) with an orthonormal basis e1, e2, e3, e4 with e2 4 := g(e4, e4) = 1. We denote by E ′ = e⊥ 4 the orthogonal complement to e4. The pseudo-sphere

Examples of Complete Solvability of 2D Classical Superintegrable Systems

Classical (maximal) superintegrable systems in $n$ dimensions are Hamiltonian systems with $2n-1$ independent constants of the motion, globally defined, the maximum number possible. They are very special because they can be solved algebraically. In this paper we show explicitly, mostly through examples of 2nd order superintegrable systems in 2 dimensions, how the trajectories can be determined in detail using rather elementary algebraic, geometric and analytic methods applied to the closed quadratic algebra of symmetries of the system. We treat a family of 2nd order degenerate systems: oscillator analogies on Darboux, nonzero constant curvature, and flat spaces, related to one another via contractions, and obeying Kepler's laws. Then we treat two 2nd order nondegenerate systems, an analogy of a caged Coulomb problem on the 2-sphere and its contraction to a Euclidean space caged Coulomb problem. In all cases the symmetry algebra structure provides detailed information about the trajectories. An interesting example is the occurrence of ''metronome orbits'', trajectories confined to an arc rather than a loop, which are indicated clearly from the structure equations but might be overlooked using more traditional methods. We also treat the Post-Winternitz system, an example of a classical 4th order superintegrable system that cannot be solved using separation of variables. Finally we treat a superintegrable system, related to the addition theorem for elliptic functions, whose constants of the motion are only rational in the momenta, a system of special interest because its constants of the motion generate a closed polynomial algebra. This paper contains many new results but we have tried to present most of the materials in a fashion that is easily accessible to nonexperts, in order to provide entr\'ee to superintegrablity theory.

Helicoidal surfaces satisfying $${\Delta ^{II}\mathbf{G}=f(\mathbf{G}+C)}$$ Δ II G = f ( G + C )

In this paper, we study helicoidal surfaces without parabolic points in Euclidean 3-space \({\mathbb{R} ^{3}}\), satisfying the condition \({\Delta ^{II}\mathbf{G}=f(\mathbf{G}+C)}\), where \({\Delta ^{II}}\) is the Laplace operator with respect to the second fundamental form, f is a smooth function on the surface and C is a constant vector. Our main results state that helicoidal surfaces without parabolic points in \({ \mathbb{R} ^{3}}\) which satisfy the condition \({\Delta ^{II} \mathbf{G}=f(\mathbf{G}+C)}\), coincide with helicoidal surfaces with non-zero constant Gaussian curvature.

Structure Relations and Darboux Contractions for 2D 2nd Order Superintegrable Systems

Two-dimensional quadratic algebras are generalizations of Lie algebras that include the symmetry algebras of 2nd order superintegrable systems in 2 dimensions as special cases. The superintegrable systems are exactly solvable physical systems in classical and quantum mechanics. Distinct superintegrable systems and their quadratic algebras can be related by geometric contractions, induced by In\"on\"u-Wigner type Lie algebra contractions. These geometric contractions have important physical and geometric meanings, such as obtaining classical phenomena as limits of quantum phenomena as ${\hbar}\to 0$ and nonrelativistic phenomena from special relativistic as $c\to \infty$, and the derivation of the Askey scheme for obtaining all hypergeometric orthogonal polynomials as limits of Racah/Wilson polynomials. In this paper we show how to simplify the structure relations for abstract nondegenerate and degenerate quadratic algebras and their contractions. In earlier papers we have classified contractions of 2nd order superintegrable systems on constant curvature spaces and have shown that all results are derivable from free quadratic algebras contained in the enveloping algebras of the Lie algebras $e(2,{\mathbb C})$ in flat space and $o(3,{\mathbb C})$ on nonzero constant curvature spaces. The quadratic algebra contractions are induced by generalizations of In\"on\"u-Wigner contractions of these Lie algebras. As a special case we obtained the Askey scheme for hypergeometric orthogonal polynomials. Here we complete this theoretical development for 2D superintegrable systems by showing that the Darboux superintegrable systems are also characterized by free quadratic algebras contained in the symmetry algebras of these spaces and that their contractions are also induced by In\"on\"u-Wigner contractions. We present tables of the contraction results.