Complete Local Classification Research Articles

Rigidity of weighted Einstein smooth metric measure spaces

We study the geometric structure of weighted Einstein smooth metric measure spaces with weighted harmonic Weyl tensor. A complete local classification is provided, showing that either the underlying manifold is Einstein, or decomposes as a warped product in a specific way. Moreover, if the manifold is complete, then it either is a weighted analogue of a space form, or it belongs to a particular family of Einstein warped products.

Killing vector fields on Riemannian and Lorentzian 3‐manifolds

AbstractWe give a complete local classification of all Riemannian 3‐manifolds admitting a nonvanishing Killing vector field T. We then extend this classification to timelike Killing vector fields on Lorentzian 3‐manifolds, which are automatically nonvanishing. The two key ingredients needed in our classification are the scalar curvature S of g and the function , where Ric is the Ricci tensor; in fact their sum appears as the Gaussian curvature of the quotient metric obtained from the action of T. Our classification generalizes that of Sasakian structures, which is the special case when . We also give necessary, and separately, sufficient conditions, both expressed in terms of , for g to be locally conformally flat. We then move from the local to the global setting, and prove two results: in the event that T has unit length and the coordinates derived in our classification are globally defined on , we give conditions under which S completely determines when the metric will be geodesically complete. In the event that the 3‐manifold M is compact, we give a condition stating when it admits a metric of constant positive sectional curvature.

Para-K\xe4hler-Einstein 4-manifolds and non-integrable twistor distributions

We study the local geometry of 4-manifolds equipped with a para-Kähler-Einstein (pKE) metric, a special type of split-signature pseudo-Riemannian metric, and their associated twistor distribution, a rank 2 distribution on the 5-dimensional total space of the circle bundle of self-dual null 2-planes. For pKE metrics with non-zero scalar curvature this twistor distribution has exactly two integral leaves and is ‘maximally non-integrable’ on their complement, a so-called (2,3,5)-distribution. Our main result establishes a simple correspondence between the anti-self-dual Weyl tensor of a pKE metric with non-zero scalar curvature and the Cartan quartic of the associated twistor distribution. This will be followed by a discussion of this correspondence for general split-signature metrics which is shown to be much more involved. We use Cartan’s method of equivalence to produce a large number of explicit examples of pKE metrics with non-zero scalar curvature whose anti-self-dual Weyl tensor have special real Petrov type. In the case of real Petrov type D, we obtain a complete local classification. Combined with the main result, this produces twistor distributions whose Cartan quartic has the same algebraic type as the Petrov type of the constructed pKE metrics. In a similar manner, one can obtain twistor distributions with Cartan quartic of arbitrary algebraic type. As a byproduct of our pKE examples we naturally obtain para-Sasaki-Einstein metrics in five dimensions. Furthermore, we study various Cartan geometries naturally associated to certain classes of pKE 4-dimensional metrics. We observe that in some geometrically distinguished cases the corresponding Cartan connections satisfy the Yang-Mills equations. We then provide explicit examples of such Yang-Mills Cartan connections.

Metric connections with parallel skew-symmetric torsion

A geometry with parallel skew-symmetric torsion is a Riemannian manifold carrying a metric connection with parallel skew-symmetric torsion. Besides the trivial case of the Levi-Civita connection, geometries with non-vanishing parallel skew-symmetric torsion arise naturally in several geometric contexts, e.g. on naturally reductive homogeneous spaces, nearly Kähler or nearly parallel G2-manifolds, Sasakian and 3-Sasakian manifolds, or twistor spaces over quaternion-Kähler manifolds with positive scalar curvature. In this paper we study the local structure of Riemannian manifolds carrying a metric connection with parallel skew-symmetric torsion. On every such manifold one can define a natural splitting of the tangent bundle which gives rise to a Riemannian submersion over a geometry with parallel skew-symmetric torsion of smaller dimension endowed with some extra structure. We show how previously known examples of geometries with parallel skew-symmetric torsion fit into this pattern, and construct several new examples. In the particular case where the above Riemannian submersion has the structure of a principal bundle, we give the complete local classification of the corresponding geometries with parallel skew-symmetric torsion.

Automatic Detection of Glaucoma based on Refined Complete Local Binary Pattern and Random Forest Classification Method

Glaucoma is considered to be one of the main root causes of blindness. As it shows no symptoms, if not properly identified at the correct time would result in the loss of vision. This paper proposes a method for the Automatic Detection of Glaucoma based on Refined Complete Local Binary Pattern and Random Forest Classification Method(RCLBP-RFC), which identifies the presence or the absence of glaucoma in a patient at an early stage. The first step is use to convert a color image into gray scale image and the second step we use Neighborhood Fuzzy K Means Clustering to segment Optic Disc(OD) and Optic Cup(OC). In the third step Statistical Optimized and Restoration model is use to extract the enhanced images using the restoration technique. In the Fourth step we exploit Refined Complete Local Binary Patterns Extraction to extract the most relevant features and finally, Random Forest Classification methods are involved to classify the features as normal, abnormal or early detected glaucoma. The experiments show that our RCLBP-RFC method achieves state-of-the-art OD and OC segmentation result on DRIONS dataset. Experimental results indicates that the proposed method identifies the presence or absence of glaucoma more precisely than other existing methods in terms of computational time and complexity, and accuracy

Hypersurfaces of space forms carrying a totally geodesic foliation

In this paper we give a complete local parametric classification of the hypersurfaces with dimension at least three of a space form that carry a totally geodesic foliation of codimension one. A classification under the assumption that the leaves of the foliation are complete was already given in Dajczer et al. (Geometriae Dedicata 176:215–224, 2015) for Euclidean hypersurfaces. We prove that there exists exactly one further class of local examples in Euclidean space, all of which have rank two. We also extend the classification under the global assumption of completeness of the leaves for hypersurfaces of the sphere and show that there exist plenty of examples in hyperbolic space.

Homogeneous Integrable Legendrian Contact Structures in Dimension Five

We consider Legendrian contact structures on odd-dimensional complex analytic manifolds. We are particularly interested in integrable structures, which can be encoded by compatible complete systems of second order PDEs on a scalar function of many independent variables and considered up to point transformations. Using the techniques of parabolic differential geometry, we compute the associated regular, normal Cartan connection and give explicit formulas for the harmonic part of the curvature. The PDE system is trivializable by means of point transformations if and only if the harmonic curvature vanishes identically. In dimension five, the harmonic curvature takes the form of a binary quartic field, so there is a Petrov classification based on its root type. We give a complete local classification of all five-dimensional integrable Legendrian contact structures whose symmetry algebra is transitive on the manifold and has at least one-dimensional isotropy algebra at any point.

On Generalized Constant Ratio Surfaces of Higher Codimension

In this paper, we introduce constant slope (CS) and generalized constant ratio (GCR) submanifolds with higher codimension in Euclidean spaces. We firstly obtain a classification of GCR surfaces in Euclidean 4-spaces $${\mathbb {E}}^4$$ . Then, we get complete local classification of CS surfaces in $${\mathbb {E}}^4$$ . We also study GCR surfaces in terms of some of its geometrical invariants.

On Para-Complex Affine Hyperspheres

In this paper we introduce a notion of a para-complex affine hypersphere. We give a complete local classification of such hypersurfaces and give several examples. It turns out that every para-complex affine hypersphere can be constructed from (real) affine hyperspheres. As an application, we classify all 2-dimensional para-complex affine hyperspheres.

The infinitesimally bendable Euclidean hypersurfaces

The main purpose of this paper is to complete the work initiated by Sbrana in 1909 giving a complete local classification of the nonflat infinitesimally bendable hypersurfaces in Euclidean space.

Lie–Hamilton systems on the plane: Properties, classification and applications

We study Lie–Hamilton systems on the plane, i.e. systems of first-order differential equations describing the integral curves of a t-dependent vector field taking values in a finite-dimensional real Lie algebra of planar Hamiltonian vector fields with respect to a Poisson structure. We start with the local classification of finite-dimensional real Lie algebras of vector fields on the plane obtained in González-López, Kamran, and Olver (1992) [23] and we interpret their results as a local classification of Lie systems. By determining which of these real Lie algebras consist of Hamiltonian vector fields relative to a Poisson structure, we provide the complete local classification of Lie–Hamilton systems on the plane. We present and study through our results new Lie–Hamilton systems of interest which are used to investigate relevant non-autonomous differential equations, e.g. we get explicit local diffeomorphisms between such systems. We also analyse biomathematical models, the Milne–Pinney equations, second-order Kummer–Schwarz equations, complex Riccati equations and Buchdahl equations.

A characterization of hyperbolic affine flat, affine minimal surfaces in [formula omitted

We investigate the geometric properties of hyperbolic affine flat, affine minimal surfaces in the equiaffine space A3. We use Cartan's method of moving frames to compute a complete set of local invariants for such surfaces. Using these invariants, we give a complete local classification of such surfaces and construct new examples.

Four-dimensional Kähler metrics admitting c-projective vector fields

A vector field on a Kähler manifold is called c-projective if its flow preserves the J-planar curves. We give a complete local classification of Kähler real 4-dimensional manifolds that admit an essential c-projective vector field. An important technical step is a local description of 4-dimensional c-projectively equivalent metrics of arbitrary signature. As an application of our results we prove the natural analog of the classical Yano–Obata conjecture in the pseudo-Riemannian 4-dimensional case.

Invariant affine connections on three-dimensional homogeneous spaces with nonsolvable transformation group

The aim of this paper is to describe all invariant affine connections on three-dimensional homogeneous spaces with nonsolvable transformation group. We present complete local classification of homogeneous spaces, it is equivalent to the description of effective pairs of Lie algebras. We describe all invariant affine connections together with their curvature and torsion tensors, holonomy algebras.

Affine connections on three-dimensional pseudo-Riemannian homogeneous spaces. II

We describe all invariant affine connections on three-dimensional pseudo-Riemannian homogeneous spaces. We present the complete local classification of pseudo-Riemannian homogeneous spaces. It is equivalent to the description of effective pairs of Lie algebras supplied with an invariant nondegenerate symmetric bilinear form on the isotropy module. We describe all invariant affine connections on pseudo-Riemannian homogeneous spaces together with their curvature and torsion tensors and single out pseudo-Riemannian connections.

The geometry of lightlike surfaces in Minkowski space

We investigate the geometric properties of lightlike surfaces in the Minkowski space R2,1, using Cartan’s method of moving frames to compute a complete set of local invariants for such surfaces. Using these invariants, we give a complete local classification of lightlike surfaces of constant type in R2,1 and construct new examples of such surfaces.

Three-dimensional conformally symmetric manifolds

The nonexistence of nontrivial conformally symmetric manifolds in the three-dimensional Riemannian setting is shown. In Lorentzian signature, a complete local classification is obtained. Furthermore, the isometry classes are examined.

Constant angle surfaces in the Heisenberg group

In this article we extend the notion of constant angle surfaces in $$ \mathbb{S}^2 $$ × ℝ and ℍ2 × ℝ to general Bianchi-Cartan-Vranceanu spaces. We show that these surfaces have constant Gaussian curvature and we give a complete local classification in the Heisenberg group.

Relative Tchebychev hypersurfaces which are also translation hypersurfaces

The class of relative Tchebychev hypersurfaces extends the class of affine hyperspheres in Blaschke geometry. This paper aims at finding examples for relative Tchebychev hypersurfaces among the translation hypersurfaces. A complete local classification is given. Specializing from relative to centroaffine geometry, the list reduces to paraboloids.

Pedal Curves Part I: Homogeneous Differential Equation

We study the class of plane curves with positive curvature κ and spherical parametrization s. t. that the curves and their derived curves like evolute, caustic, pedal and co-pedal curve, resp., have the same shape. We characterize these properties by a homogeneous linear differential equation of first order for the support function ρ and the radii of curvature κ -1 and we give a complete local classification of this class.