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1.
The equivariant real, complex and quaternionic vector fields on spheres problem is reduced to a question about the equivariant J-groups of the projective spaces. As an application of this reduction, we give a generalization of the results of Namboodiri [U. Namboodiri, Equivariant vector fields on spheres, Trans. Amer. Math. Soc. 278 (2) (1983) 431-460], on equivariant real vector fields, and Önder [T. Önder, Equivariant cross sections of complex Stiefel manifolds, Topology Appl. 109 (2001) 107-125], on equivariant complex vector fields, which avoids the restriction that the representation containing the sphere has enough orbit types.  相似文献   

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For every odd n, on the sphere S n , ρ(n) ? 1 linear orthonormal tangent vector fields, where ρ(n) is the Hurwitz-Radon number, are explicitly constructed. For each 8 × 8 sign matrix, compositions for infinite-dimensional positive definite quadratic forms are explicitly constructed. The infinite-dimensional real normed algebras thus arising are proved to have certain properties of associativity and divisibility type.  相似文献   

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In this work, a general representation for an operator self-similar Gaussian vector field is obtained, and some properties are studied. It is also shown that such a process is the operator scaling limit of a Gaussian vector field.  相似文献   

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The volume of a unit vector field V of the sphere (n odd) is the volume of its image V() in the unit tangent bundle. Unit Hopf vector fields, that is, unit vector fields that are tangent to the fibre of a Hopf fibration are well known to be critical for the volume functional. Moreover, Gluck and Ziller proved that these fields achieve the minimum of the volume if n = 3 and they opened the question of whether this result would be true for all odd dimensional spheres. It was shown to be inaccurate on spheres of radius one. Indeed, Pedersen exhibited smooth vector fields on the unit sphere with less volume than Hopf vector fields for a dimension greater than five. In this article, we consider the situation for any odd dimensional spheres, but not necessarily of radius one. We show that the stability of the Hopf field actually depends on radius, instability occurs precisely if and only if In particular, the Hopf field cannot be minimum in this range. On the contrary, for r small, a computation shows that the volume of vector fields built by Pedersen is greater than the volume of the Hopf one thus, in this case, the Hopf vector field remains a candidate to be a minimizer. We then study the asymptotic behaviour of the volume; for small r it is ruled by the first term of the Taylor expansion of the volume. We call this term the twisting of the vector field. The lower this term is, the lower the volume of the vector field is for small r. It turns out that unit Hopf vector fields are absolute minima of the twisting. This fact, together with the stability result, gives two positive arguments in favour of the Gluck and Ziller conjecture for small r.  相似文献   

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In this short note we consider an n-dimensional compact Riemannian manifold (M, g) of constant scalar curvature S = n(n − 1)c and show that the presence of a nontrivial conformal vector field ξ on M forces S to be positive. Then we show that an appropriate control on the energy of ξ makes M to be isometric to the n-sphere S n (c).  相似文献   

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In this work, we study the stability of Hopf vector fields on Lorentzian Berger spheres as critical points of the energy, the volume and the generalized energy. In order to do so, we construct a family of vector fields using the simultaneous eigenfunctions of the Laplacian and of the vertical Laplacian of the sphere. The Hessians of the functionals are negative when they act on these particular vector fields and then Hopf vector fields are unstable. Moreover, we use this technique to study some of the open problems in the Riemannian case.  相似文献   

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《Discrete Mathematics》1985,54(2):225-228
We present simple proofs of three theorems which were proved in a recent paper by J. Körner and V. Wei.  相似文献   

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We construct explicitly a maximal set of linearly independent vector fields on any odd-dimensional sphere, i.e., in the general case. The differential geometric properties of the constructed fields are studied. In particular, we find their streamlines, calculate the principal curvatures of the second kind and study holonomic properties of the distributions determined by these fields. Translated fromMatematicheskie Zametki, Vol. 67, No. 5, pp. 643–653, May, 2000.  相似文献   

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Marco Brunella 《Topology》2004,43(2):433-445
We give a full classification, up to polynomial automorphisms, of complete polynomial vector fields in two complex variables.  相似文献   

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This paper establishes a similarity principle for a class of non-elliptic, smooth complex vector fields in the plane. This principle is used to prove a uniqueness result for a nonlinear Cauchy problem.

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