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g -natural metrics of constant curvature on unit tangent sphere bundles

M. T. K. Abbassi, Giovanni Calvaruso (2012)

Archivum Mathematicum

We completely classify Riemannian g -natural metrics of constant sectional curvature on the unit tangent sphere bundle T 1 M of a Riemannian manifold ( M , g ) . Since the base manifold M turns out to be necessarily two-dimensional, weaker curvature conditions are also investigated for a Riemannian g -natural metric on the unit tangent sphere bundle of a Riemannian surface.

G -space of isotropic directions and G -spaces of ϕ -scalars with G = O ( n , 1 , ℝ )

Aleksander Misiak, Eugeniusz Stasiak (2008)

Mathematica Bohemica

There exist exactly four homomorphisms ϕ from the pseudo-orthogonal group of index one G = O ( n , 1 , ℝ ) into the group of real numbers ℝ 0 . Thus we have four G -spaces of ϕ -scalars ( ℝ , G , h ϕ ) in the geometry of the group G . The group G operates also on the sphere S n - 2 forming a G -space of isotropic directions ( S n - 2 , G , * ) . In this note, we have solved the functional equation F ( A * q 1 , A * q 2 , ⋯ , A * q m ) = ϕ ( A ) · F ( q 1 , q 2 , ⋯ , q m ) for given independent points q 1 , q 2 , ⋯ , q m ∈ S n - 2 with 1 ≤ m ≤ n and an arbitrary matrix A ∈ G considering each of all four homomorphisms. Thereby we have determined all equivariant mappings F : ( S n - 2 ) m → ℝ .

G1-structures of second order.

Demetra Demetropoulou Psomopoulou (1992)

Publicacions Matemàtiques

We introduce a generalization to the second order of the notion of the G1-structure, the so called generalized almost tangent structure. For this purpose, the concepts of the second order frame bundle H2(Vm), its structural group Lm2 and its associated tangent bundle of second order T2(Vm) of a differentiable manifold Vm are described from the point of view that is used. Then, a G1-structure of second order -called G12-structure- is constructed on Vm by an endorphism J acting on T2(Vm), satisfying...

Gap properties of harmonic maps and submanifolds

Qun Chen, Zhen Rong Zhou (2005)

Archivum Mathematicum

In this article, we obtain a gap property of energy densities of harmonic maps from a closed Riemannian manifold to a Grassmannian and then, use it to Gaussian maps of some submanifolds to get a gap property of the second fundamental forms.

Gauge equivalence of Dirac structures and symplectic groupoids

Henrique Bursztyn, Olga Radko (2003)

Annales de l’institut Fourier

We study gauge transformations of Dirac structures and the relationship between gauge and Morita equivalences of Poisson manifolds. We describe how the symplectic structure of a symplectic groupoid is affected by a gauge transformation of the Poisson structure on its identity section, and prove that gauge-equivalent integrable Poisson structures are Morita equivalent. As an example, we study certain generic sets of Poisson structures on Riemann surfaces: we find complete gauge-equivalence invariants...

Gauge theoretical methods in the classification of non-Kählerian surfaces

Andrei Teleman (2009)

Banach Center Publications

The classification of class VII surfaces is a very difficult classical problem in complex geometry. It is considered by experts to be the most important gap in the Enriques-Kodaira classification table for complex surfaces. The standard conjecture concerning this problem states that any minimal class VII surface with b₂ > 0 has b₂ curves. By the results of [Ka1]-[Ka3], [Na1]-[Na3], [DOT], [OT] this conjecture (if true) would solve the classification problem completely. We explain a new approach...

Gauge-natural field theories and Noether theorems: canonical covariant conserved currents

Palese, Marcella, Winterroth, Ekkehart (2006)

Proceedings of the 25th Winter School "Geometry and Physics"

Summary: We specialize in a new way the second Noether theorem for gauge-natural field theories by relating it to the Jacobi morphism and show that it plays a fundamental role in the derivation of canonical covariant conserved quantities. In particular we show that Bergmann-Bianchi identities for such theories hold true covariantly and canonically only along solutions of generalized gauge-natural Jacobi equations. Vice versa, all vertical parts of gauge-natural lifts of infinitesimal principal automorphisms...

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