An additive representation of the Ricci tensor on a p-surface of Euclidean space

1967 ◽  
Vol 7 (3) ◽  
pp. 404-413
Author(s):  
V. T. Bazylev
Keyword(s):  
Euclidean Space ◽  
Ricci Tensor ◽  
Additive Representation

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 ON THE GLOBAL STRUCTURE OF CONFORMAL GRADIENT SOLITONS WITH NONNEGATIVE RICCI TENSOR

2012 ◽  
Vol 14 (06) ◽  
pp. 1250045 ◽  
Author(s):  
GIOVANNI CATINO ◽  
CARLO MANTEGAZZA ◽  
LORENZO MAZZIERI
Keyword(s):  
Euclidean Space ◽  
Potential Function ◽  
Direct Product ◽  
Ricci Tensor ◽  
Global Structure ◽  
Round Sphere ◽  
Rotationally Symmetric ◽  
Gradient Soliton ◽  
Gradient Solitons

In this paper we prove that any complete conformal gradient soliton with nonnegative Ricci tensor is either isometric to a direct product ℝ × Nn-1, or globally conformally equivalent to the Euclidean space ℝn or to the round sphere 𝕊n. In particular, we show that any complete, noncompact, gradient Yamabe-type soliton with positive Ricci tensor is rotationally symmetric, whenever the potential function is nonconstant.

Symplectic capacities

2017 ◽  
Author(s):  
Dusa McDuff ◽  
Dietmar Salamon
Keyword(s):  
Euclidean Space ◽  
Generating Functions ◽  
Weinstein Conjecture ◽  
Finite Dimensional ◽  
Hofer Metric

This chapter returns to the problems which were formulated in Chapter 1, namely the Weinstein conjecture, the nonsqueezing theorem, and symplectic rigidity. These questions are all related to the existence and properties of symplectic capacities. The chapter begins by discussing some of the consequences which follow from the existence of capacities. In particular, it establishes symplectic rigidity and discusses the relation between capacities and the Hofer metric on the group of Hamiltonian symplectomorphisms. The chapter then introduces the Hofer–Zehnder capacity, and shows that its existence gives rise to a proof of the Weinstein conjecture for hypersurfaces of Euclidean space. The last section contains a proof that the Hofer–Zehnder capacity satisfies the required axioms. This proof translates the Hofer–Zehnder variational argument into the setting of (finite-dimensional) generating functions.

ANTI-SELF-DUAL SOLUTIONS OF THE YANG-MILLS EQUATIONS IN 4n DIMENSIONS

1992 ◽  
Vol 07 (23) ◽  
pp. 2077-2085 ◽  
Author(s):  
A. D. POPOV
Keyword(s):  
Scalar Field ◽  
Euclidean Space ◽  
Lie Algebra ◽  
Linear Equations ◽  
Dual Solutions ◽  
Gauge Fields ◽  
System Of Equations ◽  
Semisimple Lie Algebra ◽  
Yang Mills ◽  
Self Duality

The anti-self-duality equations for gauge fields in d = 4 and a generalization of these equations to dimension d = 4n are considered. For gauge fields with values in an arbitrary semisimple Lie algebra [Formula: see text] we introduce the ansatz which reduces the anti-self-duality equations in the Euclidean space ℝ4n to a system of equations breaking up into the well known Nahm's equations and some linear equations for scalar field φ.

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