Differential geometry of tangent bundles of order 2

The basic geometry of vector fields and definition of the notions of tangent bundles are developed in an essential different way than in usual differential geometry. ø-related vector fields are studied and some related properties are developed in our paper. Finally, a theorem 5.04 on our natural injection j of submanifolds which is j-related to vector field X is treated.DOI: http://dx.doi.org/10.3329/dujs.v60i2.11530 Dhaka Univ. J. Sci. 60(2): 259-263, 2012 (July)

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PROJECTIVE MODULE DESCRIPTION OF EMBEDDED NONCOMMUTATIVE SPACES

Reviews in Mathematical Physics ◽

10.1142/s0129055x10004028 ◽

2010 ◽

Vol 22 (05) ◽

pp. 507-531 ◽

Cited By ~ 1

Author(s):

R. B. ZHANG ◽

XIAO ZHANG

Keyword(s):

Differential Geometry ◽

Projective Modules ◽

Geometric Framework ◽

Transformation Rules ◽

Noncommutative Spaces ◽

Levi Civita Connection ◽

Coordinate Changes ◽

Moyal Algebra ◽

Tangent Bundles ◽

Algebraic Formulation

An algebraic formulation is given for the embedded noncommutative spaces over the Moyal algebra developed in a geometric framework in [8]. We explicitly construct the projective modules corresponding to the tangent bundles of the embedded noncommutative spaces, and recover from this algebraic formulation the metric, Levi–Civita connection and related curvatures, which were introduced geometrically in [8]. Transformation rules for connections and curvatures under general coordinate changes are given. A bar involution on the Moyal algebra is discovered, and its consequences on the noncommutative differential geometry are described.

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Bundle Prolongations and Connections

Gauge Field Theory in Natural Geometric Language ◽

10.1093/oso/9780198861492.003.0001 ◽

2020 ◽

pp. 3-24

Author(s):

Daniel Canarutto

Keyword(s):

Differential Geometry ◽

Covariant Derivative ◽

Vector Bundles ◽

Linear Connections ◽

Nijenhuis Bracket ◽

Fibered Manifolds ◽

Tangent Bundles ◽

Vector Valued

A synthetic introduction to the fundamental notions of differential geometry, including tangent, vertical and jet prolongations of fibered manifolds; the Frölicher-Nijenhuis bracket; connections of fibered manifolds and, in particular, linear connections of vector bundles and tangent bundles; the covariant differential of vector-valued forms as a generalisation of the standard covariant derivative.

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