scholarly journals Recurrent equiaffine projective Euclidean spaces

Filomat ◽  
2019 ◽  
Vol 33 (4) ◽  
pp. 1053-1058
Author(s):  
Almazbek Sabykanov ◽  
Josef Mikes ◽  
Patrik Peska
Keyword(s):  
Euclidean Space ◽  
Curvature Tensor ◽  
Ricci Tensor ◽  
Euclidean Spaces ◽  
Recurrent Curvature

In this paper, we study n-dimensional recurrent equiaffine projective Euclidean manifolds, i.e. manifolds with absolute recurrent curvature tensor, which admit geodesic mappings onto Euclidean space, and they are equiaffine (where was obtained the symmetric Ricci tensor). We obtained main conditions of recurrent projective Euclidean spaces and constructed their examples.

scholarly journals Characterisation of upper gradients on the weighted Euclidean space and applications

2021 ◽  
Author(s):  
Danka Lučić ◽  
Enrico Pasqualetto ◽  
Tapio Rajala
Keyword(s):  
Euclidean Space ◽  
Sobolev Space ◽  
Radon Measure ◽  
Lipschitz Functions ◽  
Euclidean Spaces ◽  
Upper Gradient

AbstractIn the context of Euclidean spaces equipped with an arbitrary Radon measure, we prove the equivalence among several different notions of Sobolev space present in the literature and we characterise the minimal weak upper gradient of all Lipschitz functions.

scholarly journals On a Set of Conform-Invariant Equations of the Gravitational Field

1953 ◽  
Vol 10 (1) ◽  
pp. 16-20 ◽  
Author(s):  
H. A. Buchdahl
Keyword(s):  
Gravitational Field ◽  
Linear Combination ◽  
Curvature Tensor ◽  
Wide Class ◽  
Ricci Tensor ◽  
Empty Space ◽  
Image Position ◽  
Derivatives Of

Eddington has considered equations of the gravitational field in empty space which are of the fourth differential order, viz. the sets of equations which express the vanishing of the Hamiltonian derivatives of certain fundamental invariants. The author has shown that a wide class of such equations are satisfied by any solution of the equationswhere Gμν and gμν are the components of the Ricci tensor and the metrical tensor respectively, whilst λ is an arbitrary constant. For a V4 this applies in particular when the invariant referred to above is chosen from the setwhere Bμνσρ is the covariant curvature tensor. K3 has been included since, according to a result due to Lanczos3, its Hamiltonian derivative is a linear combination of and , i.e. of the Hamiltonian derivatives of K1 and K2. In fact

scholarly journals Local and Nonlocal Singular Liouville Equations in Euclidean Spaces

2019 ◽  
Author(s):  
Ali Hyder ◽  
Gabriele Mancini ◽  
Luca Martinazzi
Keyword(s):  
Asymptotic Behavior ◽  
Euclidean Space ◽  
Finite Volume ◽  
Existence Of Solutions ◽  
Euclidean Spaces ◽  
Open Problems ◽  
Supercritical Regime

AbstractWe study the metrics of constant $Q$-curvature in the Euclidean space with a prescribed singularity at the origin, namely solutions to the equation \begin{equation*} (-\Delta)^{\frac{n}{2}}w=e^{nw}-c\delta_{0} \ \textrm{on}\ {\mathbb{R}}^n, \end{equation*}under a finite volume condition. We analyze the asymptotic behavior at infinity and the existence of solutions for every $n\ge 3$ also in a supercritical regime. Finally, we state some open problems.

SEQUENTIAL COLLISION-FREE OPTIMAL MOTION PLANNING ALGORITHMS IN PUNCTURED EUCLIDEAN SPACES

2020 ◽  
Vol 102 (3) ◽  
pp. 506-516
Author(s):  
CESAR A. IPANAQUE ZAPATA ◽  
JESÚS GONZÁLEZ
Keyword(s):  
Euclidean Space ◽  
Motion Planning ◽  
Euclidean Spaces ◽  
Optimal Motion ◽  
Topological Theory ◽  
Theory Of Motion ◽  
Optimal Motion Planning ◽  
Planning Algorithms

In robotics, a topological theory of motion planning was initiated by M. Farber. We present optimal motion planning algorithms which can be used in designing practical systems controlling objects moving in Euclidean space without collisions between them and avoiding obstacles. Furthermore, we present the multi-tasking version of the algorithms.

scholarly journals Chen’s Biharmonic Conjecture and Submanifolds with Parallel Normalized Mean Curvature Vector

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 710 ◽  
Author(s):  
Bang-Yen Chen
Keyword(s):  
Euclidean Space ◽  
Mean Curvature ◽  
Curvature Vector ◽  
Mean Curvature Vector ◽  
Biharmonic Submanifolds ◽  
Euclidean Spaces ◽  
Principal Curvatures ◽  
Chen’S Conjecture ◽  
Biharmonic Submanifold

The well known Chen’s conjecture on biharmonic submanifolds in Euclidean spaces states that every biharmonic submanifold in a Euclidean space is a minimal one. For hypersurfaces, we know from Chen and Jiang that the conjecture is true for biharmonic surfaces in E 3 . Also, Hasanis and Vlachos proved that biharmonic hypersurfaces in E 4 ; and Dimitric proved that biharmonic hypersurfaces in E m with at most two distinct principal curvatures. Chen and Munteanu showed that the conjecture is true for δ ( 2 ) -ideal and δ ( 3 ) -ideal hypersurfaces in E m . Further, Fu proved that the conjecture is true for hypersurfaces with three distinct principal curvatures in E m with arbitrary m. In this article, we provide another solution to the conjecture, namely, we prove that biharmonic surfaces do not exist in any Euclidean space with parallel normalized mean curvature vectors.

scholarly journals Reflection-Like Maps in High-Dimensional Euclidean Space

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 872
Author(s):  
Zhiyong Huang ◽  
Baokui Li
Keyword(s):  
Euclidean Space ◽  
Convex Domain ◽  
Orthogonal Transformation ◽  
High Dimensional ◽  
Dimensional Euclidean Space ◽  
Local Domain ◽  
Euclidean Spaces ◽  
Line Cross ◽  
Klein Model

In this paper, we introduce reflection-like maps in n-dimensional Euclidean spaces, which are affinely conjugated to θ : ( x 1 , x 2 , … , x n ) → 1 x 1 , x 2 x 1 , … , x n x 1 . We shall prove that reflection-like maps are line-to-line, cross ratios preserving on lines and quadrics preserving. The goal of this article was to consider the rigidity of line-to-line maps on the local domain of R n by using reflection-like maps. We mainly prove that a line-to-line map η on any convex domain satisfying η ∘ 2 = i d and fixing any points in a super-plane is a reflection or a reflection-like map. By considering the hyperbolic isometry in the Klein Model, we also prove that any line-to-line bijection f : D n ↦ D n is either an orthogonal transformation, or a composition of an orthogonal transformation and a reflection-like map, from which we can find that reflection-like maps are important elements and instruments to consider the rigidity of line-to-line maps.

scholarly journals Quotient manifolds of flows

2017 ◽  
Vol 26 (02) ◽  
pp. 1740005 ◽  
Author(s):  
Robert E. Gompf
Keyword(s):  
Euclidean Space ◽  
Vector Fields ◽  
Topological Manifold ◽  
Fundamental Groups ◽  
Smooth Manifolds ◽  
Euclidean Spaces ◽  
Orbit Spaces ◽  
Quotient Maps ◽  
Fixed Dimension

This paper investigates which smooth manifolds arise as quotients (orbit spaces) of flows of vector fields. Such quotient maps were already known to be surjective on fundamental groups, but this paper shows that every epimorphism of countably presented groups is induced by the quotient map of some flow, and that higher homology can also be controlled. Manifolds of fixed dimension arising as quotients of flows on Euclidean space realize all even (and some odd) intersection pairings, and all homotopy spheres of dimension at least two arise in this manner. Most Euclidean spaces of dimensions five and higher have families of topologically equivalent but smoothly inequivalent flows with quotient homeomorphic to a manifold with flexibly chosen homology. For [Formula: see text], there is a topological flow on (ℝ2r+1 − 8 points) × ℝm that is unsmoothable, although smoothable near each orbit, with quotient an unsmoothable topological manifold.

scholarly journals Finite, primitive and euclidean spaces

1988 ◽  
Vol 1 (3) ◽  
pp. 177-196 ◽  
Author(s):  
Efim Khalimsky
Keyword(s):  
Image Processing ◽  
Euclidean Space ◽  
Computer Tomography ◽  
Quotient Space ◽  
Robot Vision ◽  
Euclidean Spaces ◽  
Finite Spaces ◽  
Path Connected ◽  
Connected Spaces ◽  
Digital Spaces

Integer and digital spaces are playing a significant role in digital image processing, computer graphics, computer tomography, robot vision, and many other fields dealing with finitely or countable many objects. It is proven here that every finite T0-space is a quotient space of a subspace of some simplex, i.e. of some subspace of a Euclidean space. Thus finite and digital spaces can be considered as abstract simplicial structures of subspaces of Euclidean spaces. Primitive subspaces of finite, digital, and integer spaces are introduced. They prove to be useful in the investigation of connectedness structure, which can be represented as a poset, and also in consideration of the dimension of finite spaces. Essentially T0-spaces and finitely connected and primitively path connected spaces are discussed.

scholarly journals On endo curvature tensor of a contact metric manifold

2008 ◽  
Vol 39 (2) ◽  
pp. 177-186
Author(s):  
Mohit Kumar Dwivedi ◽  
Jae-Bok Jun ◽  
Mukut Mani Tripathi
Keyword(s):  
Curvature Tensor ◽  
Ricci Tensor ◽  
Sasakian Manifold ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Contact Metric Manifold ◽  
Necessary And Sufficient

We prove that a $ ( k ,\mu ) $-manifold with vanishing Endo curvature tensor is a Sasakian manifold. We find a necessary and sufficient condition for a non-Sasakian $ ( k ,\mu ) $-manifold %$M$ whose Endo curvature tensor $ B^{es} $ satisfies $ B^{es}(\xi ,X) \cdot S=0 $, where $S$ is the Ricci tensor. Using $ {\cal D} $-homothetic deformation we obtain an example of an $ N\left( k\right) $-contact metric manifold on which $ B^{es}(\xi ,X)\cdot S\neq 0 $.

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