Theory of microcubes

Nishimura, Hirokazu

doi:10.1007/bf02435803

Cited by 19 publications

(31 citation statements)

References 7 publications

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“…is a pullback, where the assumptive object (3,6), (4,6), (5,6), (1,7), (2, 7), (3, 7), (4, 7), (5, 7), (6, 7), (2, 4), (2,5), (3,4), (3,5)}, the assumptive mapping…”

Section: The Main Identitymentioning

confidence: 99%

“…is a pullback, where the assumptive object E [3] is (2,6), (4,6), (5,6), (1, 7), (2, 7), (3, 7), (4, 7), (5, 7), (6, 7), (1, 4),…”

Section: The Main Identitymentioning

confidence: 99%

“…url: www.acadpubl.eu within the framework of synthetic differential geometry in the previous century, for which the reader is referred to [2], [3] and [4]. This paper is devoted to the general Jacobi identity within our axiomatics of differential geometry, which will play a predominant role in a subsequent paper dealing with the Frölicher-Nijenhuis calculus.…”

Section: Introductionmentioning

confidence: 99%

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Axiomatic Differential Geometry Ii-3 -- Its Developments -- Chapter 3: The General Jacobi Identity

Nishimura¹

2013

Int. J. of Pure and Appl. Math.

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“…is a pullback, where the assumptive object (3,6), (4,6), (5,6), (1,7), (2, 7), (3, 7), (4, 7), (5, 7), (6, 7), (2, 4), (2,5), (3,4), (3,5)}, the assumptive mapping…”

Section: The Main Identitymentioning

confidence: 99%

“…is a pullback, where the assumptive object E [3] is (2,6), (4,6), (5,6), (1, 7), (2, 7), (3, 7), (4, 7), (5, 7), (6, 7), (1, 4),…”

Section: The Main Identitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Axiomatic Differential Geometry Ii-3 -- Its Developments -- Chapter 3: The General Jacobi Identity

Nishimura¹

2013

Int. J. of Pure and Appl. Math.

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“…This t is then denoted τ − τ, the strong difference of τ and τ (cf. [64], [107], [57], [68], [91]). The three last references are in the context of SDG, and the construction makes sense in the generality of micro-linear spaces, so is more general than the manifold case as discussed presently.…”

Section: The Infinitesimal (Simplicial and Cubical) Complexesmentioning

confidence: 99%

“…Microcubes are used for other purposes than differential forms in White's [107] in the context of Riemannian geometry, and in several articles by Nishimura, including [91], [94], [96].…”

Section: Microcubes and Marked Microcubesmentioning

confidence: 99%

Synthetic Geometry of Manifolds

Kock

2009

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This elegant book is sure to become the standard introduction to synthetic differential geometry. It deals with some classical spaces in differential geometry, namely 'prolongation spaces' or neighbourhoods of the diagonal. These spaces enable a natural description of some of the basic constructions in local differential geometry and, in fact, form an inviting gateway to differential geometry, and also to some differential-geometric notions that exist in algebraic geometry. The presentation conveys the real strength of this approach to differential geometry. Concepts are clarified, proofs are streamlined, and the focus on infinitesimal spaces motivates the discussion well. Some of the specific differential-geometric theories dealt with are connection theory (notably affine connections), geometric distributions, differential forms, jet bundles, differentiable groupoids, differential operators, Riemannian metrics, and harmonic maps. Ideal for graduate students and researchers wishing to familiarize themselves with the field.

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Proving the Jacobi Identity the Hard Way

Mackenzie

2012

Geometric Methods in Physics

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Abstract. Vector fields on smooth manifolds may be regarded as derivations of the algebra of smooth functions, as infinitesimal generators of flows, or as sections of the tangent bundle. The last point of view leads to a formula for the bracket which is not used very often and in terms of which such a basic matter as proving the Jacobi identity seems difficult. We present a conceptually simple proof of the Jacobi identity in terms of this formulation.Mathematics Subject Classification (2010). Primary 58A99; Secondary 18D05. Keywords. Vector fields, Jacobi identity, double vector bundles, triple vector bundles.There are three global formulas by which the bracket of vector fields can be calculated. Usually one interprets vector fields as derivations on the algebra of smooth functions, and the bracket is then the commutator of derivations. There is also the flow formula in which the bracket of vector fields is regarded as the Lie derivative of one field by the other.Thirdly, for vector fields , on a manifold , and ∈ , there is:Here :is the canonical involution. This formula involves some abuse: the RHS is a vertical vector in ( ) ( ) and therefore can be identified with an element of . For convenience we refer to (1) as the 'section formula' for the bracket.Formula (1) is much less widely used than the other two; one place in which it appears is [1, p. 297]. A proof can be extracted from [2, §3.4].By using derivations the proof of the Jacobi identity can be done in one line; a few moments experimentation with (1) may leave the reader with the impression that using the section formula is unnatural and unwieldy. The purpose of this paper is to show that there is a diagrammatic proof, very easy to visualize, starting from (1), using double and triple vector bundles. This will be important in work elsewhere -since (1) uses only the tangent functor and the canonical involution , it can be formulated in more abstract settings.

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Theory of microcubes

Cited by 19 publications

References 7 publications

Axiomatic Differential Geometry Ii-3 -- Its Developments -- Chapter 3: The General Jacobi Identity

Axiomatic Differential Geometry Ii-3 -- Its Developments -- Chapter 3: The General Jacobi Identity

Synthetic Geometry of Manifolds

Proving the Jacobi Identity the Hard Way

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