Covariant vs Contravariant Methods in Differential Geometry
英文摘要:
This is a short essay about some fundamental results on scalar curvature and the two key methods that are used to establish them.
机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。
Maung Min-Oo
This is a short essay about some fundamental results on scalar curvature and the two key methods that are used to establish them.
机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。
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We give coordinate formula and geometric description of the curvature of the tensor product connection of linear connections on vector bundles with the same base manifold. We define the covariant differential of geometric fields of certain types with respect to a pair of a linear connection on a vector bundle and a linear symmetric connection on the base manifold. We prove the generalized Bianchi identity for linear connections and we prove that the antisymmetrization of the second order covariant differential is expressed via the curvature tensors of both connections.
In this paper, we study $L^p$-boundedness ($1<p\leq 2$) of the covariant Riesz transform on differential forms for a class of non-compact weighted Riemannian manifolds without assuming conditions on derivatives of curvature. We present in particular a local version of $L^p$-boundedness of Riesz transforms under two natural conditions, namely the curvature-dimension condition, and a lower bound on the Weitzenböck curvature endomorphism. As an application, the Calderón-Zygmund inequality for $1< p\leq 2$ on weighted manifolds is derived under the curvature-dimension condition as hypothesis.
The classical Rankin-Cohen brackets are bi-differential operators from $C^\infty(\mathbb R)\times C^\infty(\mathbb R)$ into $ C^\infty(\mathbb R)$. They are covariant for the (diagonal) action of ${\rm SL}(2,\mathbb R)$ through principal series representations. We construct generalizations of these operators, replacing $\mathbb R$ by $\mathbb R^n,$ the group ${\rm SL}(2,\mathbb R)$ by the group ${\rm SO}_0(1,n+1)$ viewed as the conformal group of $\mathbb R^n,$ and functions by differential forms.
This is the original manuscript dated March 9th 1983, typeset by the Editors for the Proceedings of the Midwest Geometry Conference 2007 held in memory of Thomas Branson. Stephen Paneitz passed away on September 1st 1983 while attending a conference in Clausthal and the manuscript was never published. For more than 20 years these few pages were circulated informally. In November 2004, as a service to the mathematical community, Tom Branson added a scan of the manuscript to his website. Here we make it available more formally. It is surely one of the most cited unpublished articles. The differe…
We construct the generalized version of covariant Z_3-graded differential calculus introduced by one of us (R.K.), and then extended to the case of arbitrary Z_N grading. Here our main purpose is to establish the recurrence formulae for the N-th power of covariant q-differential D_q = d_q + A and to analyze more closely the particular case of q being an Nth primitive root of unity. The generalized notions of connection and curvature are introduced and several examples of realization are displayed for N=3 and N=4. Finally we briefly discuss the idea of infinitesimal deformations of the paramete…
$L^p$-Boundedness of the Covariant Riesz Transform on Differential Forms for $p>2$
针对完备加权黎曼流形上的微分形式,证明了协变Riesz变换在p>2时的L^p有界性,基于热核准则并应用于Calderón-Zygmund不等式。
Building on the Utiyama principle we formulate an approach to Lagrangian field theory in which exterior covariant differentials of vector-valued forms replace partial derivatives, in the sense that they take up the role played by the latter in the usual jet bundle formulation. Actually a natural Lagrangian can be written as a density on a suitable "covariant prolongation bundle"; the related momenta turn out to be natural vector-valued forms, and the field equations can be expressed in terms of covariant exterior differentials of the momenta. Currents and energy-tensors naturally also fit into…
A regular normal parabolic geometry of type $G/P$ on a manifold $M$ gives rise to sequences $D_i$ of invariant differential operators, known as the curved version of the BGG resolution. These sequences are constructed from the normal covariant derivative $\na^\om$ on the corresponding tractor bundle $V,$ where $\om$ is the normal Cartan connection. The first operator $D_0$ in the sequence is overdetermined and it is well known that $\na^\om$ yields the prolongation of this operator in the homogeneous case $M = G/P$. Our first main result is the curved version of such a prolongation. This requi…
With $\vecΔ_j\geq 0$ is the uniquely determined self-adjoint realization of the Laplace operator acting on $j$-forms on a geodesically complete Riemannian manifold $M$ and $\nabla$ the Levi-Civita covariant derivative, we prove amongst other things a Li-Yau type heat kernel bound for $\nabla \mathrm{e}^{ -t\vecΔ_j }$, if the curvature tensor of $M$ and its covariant derivative are bounded, an exponentially weighted $L^p$ bound for the heat kernel of $\nabla \mathrm{e}^{ -t\vecΔ_j }$, if the curvature tensor of $M$ and its covariant derivative are bounded, that $\nabla \mathrm{e}^{ -t\vecΔ_j }$…
We study differential cohomology on categories of globally hyperbolic Lorentzian manifolds. The Lorentzian metric allows us to define a natural transformation whose kernel generalizes Maxwell's equations and fits into a restriction of the fundamental exact sequences of differential cohomology. We consider smooth Pontryagin duals of differential cohomology groups, which are subgroups of the character groups. We prove that these groups fit into smooth duals of the fundamental exact sequences of differential cohomology and equip them with a natural presymplectic structure derived from a generaliz…
Conformally covariant differential symmetry breaking operators for a vector bundle of rank 3 over S^3
本文研究了三维球面上的三维向量丛与二维球面线丛之间微分对称破缺算子的构造与分类,确定了参数(λ, ν, m)的存在条件。
The covariant canonical formalism is a covariant extension of the traditional canonical formalism of fields. In contrast to the traditional canonical theory, it has a remarkable feature that canonical equations of gauge theories or gravity are not only manifestly Lorentz covariant but also gauge covariant or diffeomorphism covariant. A mathematical peculiarity of the covariant canonical formalism is that its canonical coordinates are differential forms on a manifold. In the present paper, we find a natural Poisson bracket of this new canonical theory, and study symplectic structure behind it. …
This paper deals with sheaves of differential operators on noncommutative algebras. The sheaves are defined by quotienting a the tensor algebra of vector fields (suitably deformed by a covariant derivative) to ensure zero curvature. As an example we can obtain enveloping algebra like relations for Hopf algebras with differential structures which are not bicovariant. Symbols of differential operators are defined, but not studied. These sheaves are shown to be in the center of as category of bimodules with flat bimodule covariant derivatives. Also holomorphic differential operators are considere…
This work further develops the properties of fractional differential forms. In particular, finite dimensional subspaces of fractional form spaces are considered. An inner product, Hodge dual, and covariant derivative are defined. Coordinate transformation rules for integral order forms are also computed. Matrix order fractional calculus is used to define matrix order forms. This is achieved by combining matrix order derivatives with exterior derivatives. Coordinate transformation rules and covariant derivative for matrix order forms are also produced. The Poincare' lemma is shown to be true fo…
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