Alan Macdonald
Using recent advances in integration theory, we give a proof of the fundamental theorem of geometric calculus. We assume only that the tangential derivative $\nabla_VF$ exists and is Lebesgue integrable. We also give sufficient conditions that $\nabla_VF$ exists.
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In this paper we give a construction of Fedosov quantization incorporating the odd variables and an analogous formula to Getzler's pseudodifferential calculus composition formula is obtained. A Fedosov type connection is constructed on the bundle of Weyl tensor Clifford algebras over the cotangent bundle of a Riemannian manifold. The quantum algebra associated with this connection is used to define a deformation of the exterior algebra of Riemannian manifolds.
We develope basic geometric quantities and properties of hypersurfaces in Carnot groups.
Basic elements of integral calculus over algebras of iterated differential forms, are presented. In particular, defining complexes for modules of integral forms are described and the corresponding berezinians and complexes of integral forms are computed. Various applications and the integral calculus over the algebra $Λ_{\infty}$ will be discussed in subsequent notes.
This article provides a pedagogically oriented introduction to geometric (Clifford) calculus on pseudo-Riemannian manifolds. Unlike usual approaches to the topic, which rely on embedding the geometric algebra either within a tensor algebra or within a vector manifold framework, here we define geometric calculus directly, by elementary methods. In particular we use an axiomatic approach that directly parallels textbook introductions to general relativity and pseudo-Riemannian geometry, so that no structure outside the metrical Clifford bundle of the manifold need be introduced. On this basis we…
We derive sub-Riemannian Ricci curvature tensor for sub-Riemannian manifolds. We provide examples including the Heisenberg group, displacement group, and Martinet sub-Riemannian structure with arbitrary weighted volumes, in which we establish analytical bounds for sub-Riemannian curvature dimension bounds and log-Sobolev inequalities. {These bounds can be used to establish the entropy dissipation results for sub-Riemannian drift diffusion processes on a compact spatial domain, in term of $L_1$ distance.} Our derivation of Ricci curvature is based on generalized Gamma $z$ calculus and $z$--Boch…
We introduce a method for evaluating integrals in geometric calculus without introducing coordinates, based on using the fundamental theorem of calculus repeatedly and cutting the resulting manifolds so as to create a boundary and allow for the existence of an antiderivative at each step. The method is a direct generalization of the usual method of integration on function of a real variable. It may lead to both practical applications and help unveil new connections to various fields of mathematics.
We introduce a proximal subdifferential and develop a calculus for nonsmooth functions defined on any Riemannian manifold $M$. We give several applications of this theory, concerning: 1) differentiability and geometrical properties of the distance function to a closed subset $C$ of $M$; 2) solvability and implicit function theorems for nonsmooth functions on $M$; 3) conditions on the existence of a circumcenter for three different points of $M$; and especially 4) fixed point theorems for expansive and nonexpansive mappings and certain perturbations of such mappings defined on $M$.
We construct and study differential and integral calculus on the space of states of a C*-algebra by equipping it with a formal smooth structure. To achieve this goal we first concentrate on the space of nonpure states of a commutative C*-algebra as a guideline for the noncommutative case. In particular, we prove Stokes' theorem over the both commutative and noncommutative smooth Wasserstein space.
Here (the last paper in a series of four) we end our presentation of the basics of a systematical approach to the differential geometry of a smooth manifold M (supporting a metric field g and a general connection del) which uses the geometric algebras of multivector and extensors (fields) developed in previous papers. The theory of the Riemann and Ricci fields of the the triple (M,del,g)is investigated to for each particular open set U of M through the introduction of a geometric structure on U, i.e., a triple (U,gamma,g) where gamma is a general connection field on U and g is a metric extenso…
In this paper we give a new proof of the (strong) displacement convexity of a class of integral functionals defined on a compact Riemannian manifold satisfying a lower Ricci curvature bound. Our approach does not rely on existence and regularity results for optimal transport maps on Riemannian manifolds, but it is based on the Eulerian point of view recently introduced by Otto-Westdickenberg and on the metric characterization of the gradient flows generated by the functionals in the Wasserstein space.
The article concerns the problem if a~given system of differential equations is identical with the Euler--Lagrange system of an~appropriate variational integral. Elementary approach is applied. The main results involve the determination of the first--order variational integrals related to the second--order Euler--Lagrange systems.
A Topological View on Integration and Exterior Calculus
本文从拓扑视角构造了任意拓扑空间上的积分与外微积分体系，证明了经典积分为其特例，并利用类似链复数的概念定义了导数，广泛应用于分形与随机分析等领域。
In this paper we review the recently proposed path-integral counterpart of the Koopman-von Neumann operatorial approach to classical Hamiltonian mechanics. We identify in particular the geometrical variables entering this formulation and show that they are essentially a basis of the cotangent bundle to the tangent bundle to phase-space. In this space we introduce an extended Poisson brackets structure which allows us to re-do all the usual Cartan calculus on symplectic manifolds via these brackets. We also briefly sketch how the Schouten-Nijenhuis, the Frölicher- Nijenhuis and the Nijenhuis-Ri…
The paper studies the harmonic maps on a direction between a Riemannian space and a generalized Lagrange space. Also, it is proved there that the solutions of C^2 class of certain ODEs or PDEs are harmonic maps, in the sense of this paper.
A Groupoid Approach to the Riemann Integral (and Path Integral Quantization of the Poisson Sigma Model)
本文提出用广群与范·埃斯特映射在紧致流形上无坐标地定义黎曼和，证明其收敛于通常积分，并探讨将其用于泊松西格玛模型的格点化路径积分量子化。
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