The Chern Connection of a $$(J^{2}=\pm 1)$$ ( J 2 = ± 1 ) -Metric Manifold of Class $${\mathcal {G}}_{1}$$ G 1

2018 ◽  
Vol 15 (4) ◽  
Author(s):  
Fernando Etayo ◽  
Araceli deFrancisco ◽  
Rafael Santamaría
Keyword(s):  
Chern Connection

ON THE GAUSS–BONNET FORMULA IN RIEMANN–FINSLER GEOMETRY

2002 ◽  
Vol 34 (3) ◽  
pp. 329-340 ◽  
Author(s):  
BRAD LACKEY
Keyword(s):  
Finsler Space ◽  
Finsler Geometry ◽  
Euler Class ◽  
Constant Volume ◽  
Finsler Spaces ◽  
Chern Connection ◽  
Civita Connection ◽  
Levi Civita Connection ◽  
Riemannian Case ◽  
Torsion Free

Using Chern's method of transgression, the Euler class of a compact orientable Riemann–Finsler space is represented by polynomials in the connection and curvature matrices of a torsion-free connection. When using the Chern connection (and hence the Christoffel–Levi–Civita connection in the Riemannian case), this result extends the Gauss–Bonnet formula of Bao and Chern to Finsler spaces whose indicatrices need not have constant volume.

PARALLEL π-VECTOR FIELDS AND ENERGY β-CHANGE

2011 ◽  
Vol 08 (04) ◽  
pp. 753-772 ◽  
Author(s):  
A. SOLEIMAN
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Finsler Geometry ◽  
Linear Connection ◽  
Free Form ◽  
Cartan Connection ◽  
Finsler Spaces ◽  
Chern Connection ◽  
Berwald Connection ◽  
Curvature Tensors

The present paper deals with an intrinsic investigation of the notion of a parallel π-vector field on the pullback bundle of a Finsler manifold (M, L). The effect of the existence of a parallel π-vector field on some important special Finsler spaces is studied. An intrinsic investigation of a particular β-change, namely the energy β-change ([Formula: see text]with[Formula: see text] being a parallel π-vector field), is established. The relation between the two Barthel connections Γ and [Formula: see text], corresponding to this change, is found. This relation, together with the fact that the Cartan and the Barthel connections have the same horizontal and vertical projectors, enable us to study the energy β-change of the fundamental linear connection in Finsler geometry: The Cartan connection, the Berwald connection, the Chern connection and the Hashiguchi connection. Moreover, the change of their curvature tensors is concluded. It should be pointed out that the present work is formulated in a prospective modern coordinate-free form.

scholarly journals Trihyperkähler reduction and instanton bundles on

2014 ◽  
Vol 150 (11) ◽  
pp. 1836-1868 ◽  
Author(s):  
Marcos Jardim ◽  
Misha Verbitsky
Keyword(s):  
Moduli Space ◽  
Twistor Space ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Rational Curves ◽  
Chern Connection ◽  
Symplectic Forms ◽  
Complex Dimension ◽  
Instanton Bundles ◽  
Hyperkähler Manifold

AbstractA trisymplectic structure on a complex $2n$-manifold is a three-dimensional space ${\rm\Omega}$ of closed holomorphic forms such that any element of ${\rm\Omega}$ has constant rank $2n$, $n$ or zero, and degenerate forms in ${\rm\Omega}$ belong to a non-degenerate quadric hypersurface. We show that a trisymplectic manifold is equipped with a holomorphic 3-web and the Chern connection of this 3-web is holomorphic, torsion-free, and preserves the three symplectic forms. We construct a trisymplectic structure on the moduli of regular rational curves in the twistor space of a hyperkähler manifold, and define a trisymplectic reduction of a trisymplectic manifold, which is a complexified form of a hyperkähler reduction. We prove that the trisymplectic reduction in the space of regular rational curves on the twistor space of a hyperkähler manifold $M$ is compatible with the hyperkähler reduction on $M$. As an application of these geometric ideas, we consider the ADHM construction of instantons and show that the moduli space of rank $r$, charge $c$ framed instanton bundles on $\mathbb{C}\mathbb{P}^{3}$ is a smooth trisymplectic manifold of complex dimension $4rc$. In particular, it follows that the moduli space of rank two, charge $c$ instanton bundles on $\mathbb{C}\mathbb{P}^{3}$ is a smooth complex manifold dimension $8c-3$, thus settling part of a 30-year-old conjecture.

ENERGY β-CONFORMAL CHANGE IN FINSLER GEOMETRY

2012 ◽  
Vol 09 (04) ◽  
pp. 1250029 ◽  
Author(s):  
A. SOLEIMAN
Keyword(s):  
Vector Field ◽  
Finsler Geometry ◽  
Linear Connection ◽  
Finsler Manifold ◽  
Free Form ◽  
Cartan Connection ◽  
Chern Connection ◽  
Conformal Change ◽  
Berwald Connection ◽  
Curvature Tensors

The present paper deals with an intrinsic generalization of the conformal change and energy β-change on a Finsler manifold (M.L.), namely the energy β-conformal change ([Formula: see text] with [Formula: see text]; [Formula: see text] being a concurrent π-vector field and σ(x) is a function on M). The relation between the two Barthel connections Γ and [Formula: see text], corresponding to this change, is found. This relation, together with the fact that the Cartan and the Barthel connections have the same horizontal and vertical projectors, enable us to study the energy β-conformal change of the fundamental linear connection in Finsler geometry: the Cartan connection, the Berwald connection, the Chern connection and the Hashiguchi connection. Moreover, the change of their curvature tensors is obtained. It should be pointed out that the present work is formulated in a prospective modern coordinate-free form.

PROLONGEMENT D’UN FIBRE HOLOMORPHE HERMITIEN A COURBURE Lp SUR UNE COURBE OUVERTE

1992 ◽  
Vol 03 (04) ◽  
pp. 441-453 ◽  
Author(s):  
OLIVIER BIQUARD
Keyword(s):  
Riemann Surface ◽  
Vector Bundle ◽  
Weighted Sobolev Space ◽  
Holomorphic Vector Bundle ◽  
Chern Connection ◽  
Hermitian Metrics ◽  
Parabolic Bundle ◽  
Finite Set ◽  
Marked Points

Let X be a Riemann surface and S a finite set of marked points on X. If [Formula: see text] is a hermitian holomorphic vector bundle with Lp-curvature for some p>1, we study the asymptotic behaviour of the Chern connection around the marked points; by solving directly a [Formula: see text]-problem in a weighted Sobolev space, we extend the holomorphic structure of [Formula: see text] over S to get a parabolic bundle. We deduce a proof of the classification of these hermitian metrics.

Sign in / Sign up

Export Citation Format

Share Document