Geodesic Spheres Research Articles

The stability of hypersurfaces revisited

In this paper, we revisit the problem of characterize (r, s)-stable closed hypersurfaces immersed in a Riemannian space form, which was firstly established in Velasquez et al. (J Math Anal Appl 406:134–146, 2013). With a different approach of that used in the proof of the main theorem of Velasquez et al. (J Math Anal Appl 406:134–146, 2013), we complete its program showing that a closed hypersurface contained in the Euclidian space \({\mathbb {R}}^{n+1}\) and having higher order mean curvatures linearly related is (r, s)-stable if, and only if, it is a geodesic sphere of \({\mathbb {R}}^{n+1}\).

Harmonic functions on Manifolds whose large spheres are small.

We study the growth of harmonic functions on complete Riemann-ian manifolds where the extrinsic diameter of geodesic spheres is sublinear. It is an generalization of a result of A. Kazue. We also get a Cheng and Yau estimates for the gradient of harmonic functions.

A note on Weingarten hypersurfaces in the warped product manifold

In this paper, we consider the closed embedded hypersurface Σ in the warped product manifold [Formula: see text] equipped with the metric g = dr2 + λ(r)2 gN. We give some characterizations of slice {r} × N by the condition that Σ has constant weighted higher-order mean curvatures (λ′)αpk, or constant weighted higher-order mean curvature ratio (λ′)αpk/p1, which generalize Brendle's [Constant mean curvature surfaces in warped product manifolds, Publ. Math. Inst. Hautes Études Sci. 117 (2013) 247–269] and Brendle–Eichmair's [Isoperimetric and Weingarten surfaces in the Schwarzschild manifold, J. Differential Geom. 94(3) (2013) 387–407] results. In particular, we show that the assumption convex of Brendle–Eichmair's result [Isoperimetric and Weingarten surfaces in the Schwarzschild manifold, J. Differential Geom. 94(3) (2013) 387–407] is unnecessary. Here pk is the kth normalized mean curvature of the hypersurface Σ. As a special case, we also give some characterizations of geodesic spheres in ℝn, ℍn and [Formula: see text], which generalize the classical Alexandrov-type results.

The Mass of Graviton and Its Relation to the Number of Information according to the Holographic Principle.

We investigate the relation of the mass of the graviton to the number of information N in a flat universe. As a result we find that the mass of the graviton scales as . Furthermore, we find that the number of gravitons contained inside the observable horizon is directly proportional to the number of information N; that is, N gr ∝ N. Similarly, the total mass of gravitons that exist in the universe is proportional to the number of information N; that is, . In an effort to establish a relation between the graviton mass and the basic parameters of the universe, we find that the mass of the graviton is simply twice the Hubble mass m H as it is defined by Gerstein et al. (2003), times the square root of the quantity q − 1/2, where q is the deceleration parameter of the universe. In relation to the geometry of the universe we find that the mass of the graviton varies according to the relation , and therefore m gr obviously controls the geometry of the space time through a deviation of the geodesic spheres from the spheres of Euclidean metric.

A NEW RESULT ABOUT ALMOST UMBILICAL HYPERSURFACES OF REAL SPACE FORMS

AbstractIn this short note, we prove that an almost umbilical compact hypersurface of a real space form with almost Codazzi umbilicity tensor is embedded, diffeomorphic and quasi-isometric to a round sphere. Then, we derive a new characterisation of geodesic spheres in space forms.

Foliations of a smooth metric measure space by hypersurfaces with constantf-mean curvature

We study smooth codimension-one foliations F of a smooth metric measure space whose leaves have the same constant f -mean curvature. Firstly, we show that all the leaves of F are f -minimal hypersurfaces when either the smooth metric measure space is compact and has nonnegative Bakry‐ Emery Ricci curvature, or the limit of the ratio of the weighted volume of a geodesic ball B and the weighted area of a geodesic sphere @B vanishes. Secondly, we prove that every leaf of F is strongly f -stable. Lastly, we show that there is no complete proper foliation of the Gaussian space whose leaves have the same constant f -mean curvature. In particular, there are no foliations of R nC1 whose leaves are complete proper self-similar solutions for mean curvature flow.

Algebraic invariants, mutation, and commensurability of link complements

We construct a family of hyperbolic link complements by gluing tangles along totally geodesic four-punctured spheres, then investigate the commensurability relation among its members. Those with different volume are incommensurable, distinguished by their scissors congruence classes. Mutation produces arbitrarily large finite subfamilies of nonisometric manifolds with the same volume and scissors congruence class. Depending on the choice of mutation, these manifolds may be commensurable or incommensurable, distinguished in the latter case by cusp parameters. All have trace field Q(i,\sqrt{2}), but some have integral traces while others do not.

A note on magnetic curves on [formula omitted

We prove that a normal magnetic curve on the Sasakian sphere S2n+1 lies on a totally geodesic sphere S3, and that the Sasakian structure on S3 is that induced from S2n+1.

Geodesic Domes and its Applications

A man made structure becomes stronger as its size is increased. They find application in the roofing of buildings. Geodesic spheres are arrangements of polygons in great circles that approximate a true sphere. Of all structures built from linear elements, a Geodesic Dome has the highest ratio of enclosed volume to weight. This paper highlights the uses and applications of geodesic domes along with their geometrical properties.

Geodesic Domes and its Applications

A man made structure becomes stronger as its size is increased. They find application in the roofing of buildings. Geodesic spheres are arrangements of polygons in great circles that approximate a true sphere. Of all structures built from linear elements, a Geodesic Dome has the highest ratio of enclosed volume to weight. This paper highlights the uses and applications of geodesic domes along with their geometrical properties.

Local two-radii theorems on the multi-dimensional sphere

Consider those functions on the -dimensional sphere that have zero integrals over all geodesic balls with centres in a given set . We obtain a description of such functions in the case when is a geodesic sphere on . We also find a criterion for the existence of non-zero functions with this property in the case when the set of centres is the union of two geodesic spheres. We obtain analogues of these results for quasi-analytic classes of functions.

Lagrangian Floer homology of a pair of real forms in Hermitian symmetric spaces of compact type

In this paper we calculate the Lagrangian Floer homology $HF(L_0, L_1 : {\mathbb Z}_2)$ of a pair of real forms $(L_0,L_1)$ in a monotone Hermitian symmetric space $M$ of compact type in the case where $L_0$ is not necessarily congruent to $L_1$. In particular, we have a generalization of the Arnold-Givental inequality in the case where $M$ is irreducible. As its application, we prove that the totally geodesic Lagrangian sphere in the complex hyperquadric is globally volume minimizing under Hamiltonian deformations.

A note on hypersurfaces in a sphere

We study the influence of a unit Killing vector field on the geometry of a hypersurface in the unit sphere. The combination of the Killing vector field on the hypersurface and the conformal vector field on the ambient sphere triggers the presence of four specific smooth functions on the hypersurface, we use these four functions to derive different sufficient conditions for a hypersurface to be the totally geodesic sphere and for a minimal hypersurface to be the totally geodesic sphere, Clifford minimal hypersurface respectively. In particular we classify compact minimal hypersurfaces with a unit Killing vector field in the unit sphere.

Globally optimal rotation alignment of spherical surfaces with associated scalar values

We propose a new approach to global optimization algorithm based on controlled random search techniques for rotational alignment of spherical surfaces with associated scalar values. To reduce the distortion in correspondence and increase efficiency, the spherical surface is first re-sampled using a geodesic sphere. The rotation in space is represented using the modified Rodrigues parameters. Correspondence between two spherical surfaces is implemented in the parametric domain. We applied the methods to the alignment of beam patterns computed from the outer ear shapes of bats. The proposed method is compared with other approaches such as principal component analysis (PCA), exhaustive search in the discrete space of rotations defined by Euler angles and direct search using uniform samples over the special orthogonal group of rotations in 3D space. Experimental results demonstrate that the rotation alignment obtained using the proposed algorithm has a high degree of precision and gives the best results among the four approaches. Open image in new window

On the stability of hypersurfaces in space forms

We define the notion of (r,s)-stability concerning closed hypersurfaces with higher order mean curvatures linearly related in a Riemannian space form. By supposing that such a hypersurface Mn is contained either in an open hemisphere of the Euclidean sphere or in the hyperbolic space, we are able to show that Mn is (r,s)-stable if, and only if, Mn is a geodesic sphere. Moreover, we obtain a suitable characterization of the (r,s)-stability through the analysis of the first eigenvalue of the Jacobi operator associated to the corresponding variational problem.

CM-BOF: visual similarity-based 3D shape retrieval using Clock Matching and Bag-of-Features

Content-based 3D object retrieval has become an active topic in many research communities. In this paper, we propose a novel visual similarity-based 3D shape retrieval method (CM-BOF) using Clock Matching and Bag-of-Features. Specifically, pose normalization is first applied to each object to generate its canonical pose, and then the normalized object is represented by a set of depth-buffer images captured on the vertices of a given geodesic sphere. Afterwards, each image is described as a word histogram obtained by the vector quantization of the image's salient local features. Finally, an efficient multi-view shape matching scheme (i.e., Clock Matching) is employed to measure the dissimilarity between two models. When applying the CM-BOF method in non-rigid 3D shape retrieval, multidimensional scaling (MDS) should be utilized before pose normalization to calculate the canonical form for each object. This paper also investigates several critical issues for the CM-BOF method, including the influence of the number of views, codebook, training data, and distance function. Experimental results on five commonly used benchmarks demonstrate that: (1) In contrast to the traditional Bag-of-Features, the time-consuming clustering is not necessary for the codebook construction of the CM-BOF approach; (2) Our methods are superior or comparable to the state of the art in applications of both rigid and non-rigid 3D shape retrieval.

Real hypersurfaces of a complex space form

In this paper we are interested in obtaining a condition under which a compact real hypersurface of a complex projective space CP n is a geodesic sphere. We also study the question as to whether the characteristic vector field of a real hypersurface of the complex projective space CP n is harmonic, and show that the answer is in negative.

Method of analytic continuation for the inverse spherical mean transform in constant curvature spaces

The following problem arises in thermoacoustic tomography and has intimate connection with PDEs and integral geometry. Reconstruct a function f supported in an n-dimensional ball B given the spherical means of f over all geodesic spheres centered on the boundary of B. We propose a new approach to this problem, which yields explicit reconstruction formulas in arbitrary constant curvature space, including euclidean space ℝ n , the n-dimensional sphere, and hyperbolic space. The main idea is analytic continuation of the corresponding operator families. The results are applied to inverse problems for a large class of Euler-Poisson-Darboux equations in constant curvature spaces of arbitrary dimension.

On the rigidity of hypersurfaces into space forms

Our purpose is to study the rigidity of complete hypersurfaces immersed into a Riemannian space form. In this setting, first we use a classical characterization of the Euclidean sphere \(\mathbb S ^{n+1}\) due to Obata (J Math Soc Jpn 14:333–340, 1962) in order to prove that a closed orientable hypersurface \(\Sigma ^n\) immersed with null second-order mean curvature in \(\mathbb S ^{n+1}\) must be isometric to a totally geodesic sphere \(\mathbb S ^{n}\), provided that its Gauss mapping is contained in a closed hemisphere. Furthermore, as suitable applications of a maximum principle at the infinity for complete noncompact Riemannian manifolds due to Yau (Indiana Univ Math J 25:659–670, 1976), we establish new characterizations of totally geodesic hypersurfaces in the Euclidean and hyperbolic spaces. We also obtain a lower estimate of the index of minimum relative nullity concerning complete noncompact hypersurfaces immersed in such ambient spaces.

D’Atri Spaces of Type k and Related Classes of Geometries Concerning Jacobi Operators

In this article we continue the study of the geometry of $k$-D'Atri spaces, $% 1\leq k$ $\leq n-1$ ($n$ denotes the dimension of the manifold)$,$ began by the second author. It is known that $k$-D'Atri spaces, $k\geq 1,$ are related to properties of Jacobi operators $R_{v}$ along geodesics, since she has shown that ${\operatorname{tr}}R_{v}$, ${\operatorname{tr}}R_{v}^{2}$ are invariant under the geodesic flow for any unit tangent vector $v$. Here, assuming that the Riemannian manifold is a D'Atri space, we prove in our main result that ${\operatorname{tr}}R_{v}^{3}$ is also invariant under the geodesic flow if $ k\geq 3$. In addition, other properties of Jacobi operators related to the Ledger conditions are obtained and they are used to give applications to Iwasawa type spaces. In the class of D'Atri spaces of Iwasawa type, we show two different characterizations of the symmetric spaces of noncompact type: they are exactly the $\frak{C}$-spaces and on the other hand they are $k$ -D'Atri spaces for some $k\geq 3.$ In the last case, they are $k$-D'Atri for all $k=1,...,n-1$ as well. In particular, Damek-Ricci spaces that are $k$-D'Atri for some $k\geq 3$ are symmetric. Finally, we characterize $k$-D'Atri spaces for all $k=1,...,n-1$ as the $% \frak{SC}$-spaces (geodesic symmetries preserve the principal curvatures of small geodesic spheres). Moreover, applying this result in the case of 4% -dimensional homogeneous spaces we prove that the properties of being a D'Atri (1-D'Atri) space, or a 3-D'Atri space, are equivalent to the property of being a $k$-D'Atri space for all $k=1,2,3$.