Multivector and Extensor Fields on Smooth Manifolds

Geometric and Extensor Algebras and the Differential Geometry of Arbitrary Manifolds

We give in this paper which is the third in a series of four a theory of covariant derivatives of representatives of multivector and extensor fields on an arbitrary open set U of M, based on the geometric and extensor calculus on an arbitrary smooth manifold M. This is done by introducing the notion of a connection extensor field gamma defining a parallelism structure on U, which represents in a well defined way the action on U of the restriction there of some given connection del defined on M. Also we give a novel and intrinsic presentation (i.e., one that does not depend on a chosen orthonor…

Extensor in Geometric Algebras

This paper, the third in a series of eight introduces some of the basic concepts of the theory of extensors needed for our formulation of the differential geometry of smooth manifolds . Key notions such as the extension and generalization operators of a given linear operator (a (1,1)-extensor) acting on a real vector space V are introduced and studied in details. Also, we introduce the notion of the determinant of a (1,1)-extensor and the concepts of standard and metric Hodge (star) operators, disclosing a non trivial and useful relation between them.

Metric Compatible Covariant Derivatives

This paper, sixth in a series of eight, uses the geometric calculus on manifolds developed in previous papers of the series to introduce through the concept of a metric extensor field g a metric structure for a smooth manifold M. The associated Christoffel operators, a notable decomposition of that object and the associated Levi-Civita connection field are given. The paper introduces also the concept of a geometrical structure for a manifold M as a triple (M,g,gamma), where gamma is a connection extensor field defining a parallelism structure for M . Next, the theory of metric compatible covar…

Parallelism Structure on a Smooth Manifold

Using the theory of extensors developed in a previous paper we present a theory of the parallelism structure on arbitrary smooth manifold. Two kinds of Cartan connection operators are introduced and both appear in intrinsic versions (i.e., frame independent) of the first and second Cartan structure equations. Also, the concept of deformed parallelism structures and relative parallelism structures which play important role in the understanding of geometrical theories of the gravitational field are investigated.

无限维中的Schouten-Nijenhuis括号

The Schouten-Nijenhuis bracket in infinite dimensions

本文在无限维流形上分两步构造Schouten-Nijenhuis括号，适用于可求和多向量场及其扩展形式，需满足分离点条件或使用局部截面层。

Differential complexes and exterior calculus

In this paper we present a new theory of calculus over $k$-dimensional domains in a smooth $n$-manifold, unifying the discrete, exterior, and continuum theories. The calculus begins at a single point and is extended to chains of finitely many points by linearity, or superposition. It converges to the smooth continuum with respect to a norm on the space of ``pointed chains,'' culminating in the chainlet complex. Through this complex, we discover a broad theory of coordinate free, multivector analysis in smooth manifolds for which both the classical Newtonian calculus and the Cartan exterior cal…

双李代数胚的外尔代数

The Weil algebra of a double Lie algebroid

针对双向量丛$D\to M$定义双分次外尔代数$\mathcal{W}(D)$，揭示其与对偶双向量丛外尔代数的关系，等价描述双线性泊松结构，给出双李代数胚结构刻画的新证明，并统一解释李代数胚形变复形等相关理论。