Chapter I The Differential Geometry of Higher Order Jets and Tangent Bundles

1985 ◽  
pp. 1-72
Keyword(s):  
Differential Geometry ◽  
Higher Order ◽  
Tangent Bundles

scholarly journals Generalizations and applications of Young’s integral inequality by higher order derivatives

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Jun-Qing Wang ◽  
Bai-Ni Guo ◽  
Feng Qi
Keyword(s):  
Differential Geometry ◽  
Integral Inequality ◽  
Higher Order ◽  
Integral Inequalities ◽  
List Type ◽  
Definite Integral ◽  
Exponential Integral ◽  
Definite Integrals ◽  
Logarithmic Integral

Abstract In the paper, the authors generalize Young’s integral inequality via Taylor’s theorems in terms of higher order derivatives and their norms, and apply newly-established integral inequalities to estimate several concrete definite integrals, including a definite integral of a function which plays an indispensable role in differential geometry and has a connection with the Lah numbers in combinatorics, the exponential integral, and the logarithmic integral.

scholarly journals The extension of G-foliations to Tangent bundles of higher order

1975 ◽  
Vol 56 ◽  
pp. 29-44 ◽  
Author(s):  
Luis A. Cordero
Keyword(s):  
Infinite Sequence ◽  
Differentiable Manifold ◽  
Higher Order ◽  
Characteristic Classes ◽  
Tangent Bundles

In this paper we describe a canonical procedure for constructing the extension of a G-foliation on a differentiable manifold X to its tangent bundles of higher order and by applying the Bott-Haefliger’s construction of characteristic classes of G-foliations ([2], [3]) we obtain an infinite sequence of characteristic classes for those foliations (Theorem 4.8).

scholarly journals Differential geometry of tangent bundles of order 2

1968 ◽  
Vol 20 (3) ◽  
pp. 318-354 ◽  
Author(s):  
Kentaro Yano ◽  
Shigeru Ishihara
Keyword(s):  
Differential Geometry ◽  
Tangent Bundles

scholarly journals Higher-Order Geodesic Equations from Non-Local Lagrangians and Complex Backward-Forward Derivative Operators

2016 ◽  
Vol 54 (1) ◽  
pp. 139-157
Author(s):  
Rami Ahmad El-Nabulsi
Keyword(s):  
Differential Geometry ◽  
Higher Order ◽  
Order Derivative ◽  
Lagrangian Dynamics ◽  
Derivative Operator ◽  
Geodesic Equations ◽  
Non Local ◽  
Extended Complex

Abstract Starting with an extended complex backwardforward derivative operator in differential geometry which is motivated from non-local-in-time Lagrangian dynamics, higher-order geodesic equations are obtained within classical differential geometrical settings. We limit our analysis up to the 2nd-order derivative where some applications are discussed and a number of features are revealed accordingly.

scholarly journals Liftings of Some Types of Tensor Fields and Connections to Tangent Bundles of pr-Velocities

1970 ◽  
Vol 40 ◽  
pp. 13-31 ◽  
Author(s):  
Akihiko Morimoto
Keyword(s):  
Vector Fields ◽  
Higher Order ◽  
Tensor Fields ◽  
Multi Index ◽  
Differentiable Functions ◽  
Tangent Bundles

In the previous paper [6], we studied the liftings of tensor fields to tangent bundles of higher order. The purpose of the present paper is to generalize the results of [6] to the tangent bundles of pr-velocities in a manifold M— notions due to C. Ehresmann [1] (see also [2]). In §1, we explain the pr-velocities in a manifold and define the (Λ)-lifting of differentiable functions for any multi-index λ -(λ1, λ2,…,λp) of non-negative integers λi satisfying ΣΛi≤r. In § 2, we construct ‹Λ›-lifts of any vector fields and ‹Λ›-lifts of 1-forms. The ‹Λ›-lift is a little bit different from the ‹Λ›-lift of vector fields in [6].

scholarly journals Prolongations of Connections to Tangential Fibre Bundles of Higher Order

1970 ◽  
Vol 40 ◽  
pp. 85-97 ◽  
Author(s):  
Akihiko Morimoto
Keyword(s):  
Tangent Bundle ◽  
Higher Order ◽  
Fibre Bundles ◽  
Affine Connections ◽  
Tangent Bundles

In the previous paper [3] we have studied the prolongations of G-structures to tangent bundles of higher order. The purpose of the present paper is to study the prolongations of connections to tangential fibre bundles of higher order, and to generalize the results due to S. Kobayashi [1] for the case of usual tangent bundle —— in fact, the arguments in [1] will be, in a sense, more or less simplified and clarified by using the notion of tangent bundles of higher order. In addition, as a consequence of our results, we shall obtain the prolongations of linear (affine) connections to tangent bundles of higher order.

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