The structure of algebraic covariant derivative curvature tensors

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…

On a new normalization for tractor covariant derivatives

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…

k阶协变导数的公式

A formula for the k-th covariant derivative

本文引入依赖于克里斯托费尔符号的P^{i<k>}_{j}和Q^{i<k>}_{j}符号，给出张量场沿曲线的k阶协变导数公式(3.1)并推广至k∈R，解决k阶协变导数的表达式问题。

Covariant Derivatives of Mutivector and Extensor Fields

We give in this paper which is the fifth in a series of eight a theory of covariant derivatives of multivector and extensor fields based on the geometric calculus of an arbitrary smooth manifold M, and the notion of a connection extensor field defining a parallelism structure on M. Also we give a novel and intrinsic presentation (i.e., one that does not depend on a chosen orthonormal moving frame) of the torsion and curvature fields of Cartan's theory. Two kinds of Cartan's connection fields are identified, and both appear in the intrinsic Cartan's structure equations satisfied by the Cartan's…

Covariant derivative of the curvature tensor of pseudo-Kählerian manifolds

It is well known that the curvature tensor of a pseudo-Riemannian manifold can be decomposed with respect to the pseudo-orthogonal group into the sum of the Weyl conformal curvature tensor, the traceless part of the Ricci tensor and of the scalar curvature. A similar decomposition with respect to the pseudo-unitary group exists on a pseudo-Kählerian manifold; instead of the Weyl tensor one obtains the Bochner tensor. In the present paper, the known decomposition with respect to the pseudo-orthogonal group of the covariant derivative of the curvature tensor of a pseudo-Riemannian manifold is re…

Linear Independence of Covariant Derivatives and Space-Curvatures

It is developed the considerations from (S. M. Minčić, [14, 15]) about curvature tensors and pseudotensors for a non-symmetric affine connection space in this paper. How many kinds of covariant derivatives are enough to be defined for complete researching in the field of non-symmetric affine connection spaces is examined here. This result is interpreted. After that, we searched how many curvature tensors and pseudotensors of a non-symmetric affine connection space are linearly independent. In the next, it is examined is it possible to express all curvature tensors as the linear combinations of…

Covariant Derivatives of Multivector and Multiform Fields

A simple theory of the covariant derivatives, deformed derivatives and relative covariant derivatives of multivector and multiform fields is presented using algebraic and analytical tools developed in previous papers.

伪黎曼子流形的测地法坐标与自然张量

Geodesic normal coordinates and natural tensors for pseudo-Riemannian submanifolds

本文构造了适用于伪黎曼子流形的测地法坐标，证明度量泰勒系数可由背景曲率张量与第二基本形式的协变导数的通用多项式表示，并据此定义和刻画了自然子流形张量与微分算子。

目标导数作为黎曼度量流形上的协变导数

Objective rates as covariant derivatives on the manifold of Riemannian metrics

本文证明在自然假设下，连续介质力学中的所有客观导数均等价于黎曼度量流形上的协变导数，并利用莱布尼茨法则统一了李型与共旋型区分，为文献中的客观导数提供了对应协变导数。

Commutation relations on the covariant derivative

The n-th order covariant derivative on a smooth manifold with an affine connection is a differential operator which turns a function into a tensor field of type (0,n). In this paper the properties of this operatior related to the permutation of indices are investigated by means of non-associative algebra. The general formula for commutation relations of this kind is obtained.

Covariant derivatives for Ehresmann connections

We deal with the construction of covariant derivatives for some quite general Ehresmann connections on fibre bundles. We show how the introduction of a vertical endomorphism allows construction of covariant derivatives separately on both the vertical and horizontal distributions of the connection which can then be glued together on the total space. We give applications across an important class of tangent bundle cases, frame bundles and the Hopf bundle.

Covariant derivatives of eigenfunctions along parallel tensors over space forms and a conjecture motivated by the vertex algebraic structure

We study the covariant derivatives of an eigenfunction for the Laplace-Beltrami operator on a complete, connected Riemannian manifold with nonzero constant sectional curvature. We show that along every parallel tensor, the covariant derivative is a scalar multiple of the eigenfunction. We also show that the scalar is a polynomial depending on the eigenvalue and prove some properties. A conjecture motivated by the study of vertex algebraic structure on space forms is also announced, suggesting the existence of interesting structures in these polynomials that awaits further exploration.

Estimates for the covariant derivative of the heat semigroup on differential forms, and covariant Riesz transforms

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 $\nabla \mathrm{e}^{ -t\vecΔ_j }$…

更高协变导数与Dirac电流丛

Higher Covariant Derivative and the Bundle of Dirac Currents

本文通过定义更高阶协变导数，构建了Dirac电流丛与广义Weyl代数之间的映射，揭示了协变导数与Riemann曲率张量的关系，并探讨了其在微分余代数中的应用。

表示值Bott-Shulman-Stasheff复形与Weil复形中的协变导数

Covariant derivatives in the representation-valued Bott-Shulman-Stasheff and Weil complex

本文针对李群胚表示值形式的双复形构造问题，提出两类与单纯微分交换的协变导数，重点研究基于理想丛乘性Ehresmann联络的水平外协变导数$D$，并探讨其在李代数胚中的对应及与van Est映射的关系，最终用于研究乘性Ehresmann联络的曲率。

具有类光或零协变导数的类时扭量空间的截面

Sections of time-like twistor spaces with light-like or zero covariant derivatives

本研究围绕类时扭量空间中具有类光或零协变导数的截面展开，分析了中性流形上h反转殆仿复结构的性质，构造了相关例子并给出E^4_2上的所有对应结构对。

Covariant Time Derivatives for Dynamical Systems

We present a unified derivation of covariant time derivatives, which transform as tensors under a time-dependent coordinate change. Such derivatives are essential for formulating physical laws in a frame-independent manner. Three specific derivatives are described: convective, corotational, and directional. The covariance is made explicit by working in arbitrary time-dependent coordinates, instead of restricting to Eulerian (fixed) or Lagrangian (material) coordinates. The commutator of covariant time and space derivatives is interpreted in terms of a time-curvature that shares many properties…

（弯曲）时空中的随机协变导数：Fractoid 空间一瞥

Stochastic Covariant Derivatives in a (Curved) Space-Time: a Glimpse into the Fractoid Spaces

本文基于 Itô-Wiener 过程与尺度相对论，系统分析平坦及弯曲时空中的随机协变导数及其相关算子、方程、积分、随机测地线和挠率，并引出 fractoid 空间。

Counterexamples to a divergence lower bound for the covariant derivative of skew-symmetric 2-tensor fields

In arXiv:2103.15482 an estimate for suitable skew-symmetric 2-tensors was claimed. Soon after, this estimate has been exploited to claim powerful classification results: most notably, it has been employed to propose a proof of a Black Hole Uniqueness Theorem for vacuum static spacetimes with positive scalar curvature and in connection with the Besse Conjecture. In the present note we point out an issue in the argument proposed in arXiv:2103.15482 and we provide a counterexample to the estimate.

Short formulas for algebraic covariant derivative curvature tensors via Algebraic Combinatorics

We consider generators of algebraic covariant derivative curvature tensors R' which can be constructed by a Young symmetrization of product tensors W*U or U*W, where W and U are covariant tensors of order 2 and 3. W is a symmetric or alternating tensor whereas U belongs to a class of the infinite set S of irreducible symmetry classes characterized by the partition (2,1). Using Computer Algebra we search for such generators whose coordinate representations are polynomials with a minimal number of summands. For a generic choice of the symmetry class of U we obtain lengths of 16 or 20 summands if…

Geodesic mapping onto Kählerian space of the first kind

In the present paper a generalized Kählerian space $\mathbb{G}\underset 1 {\mathbb{K}}{}_N$ of the first kind is considered, as a generalized Riemannian space $\mathbb{GR}_N$ with almost complex structure $F^h_i$, that is covariantly constant with respect to the first kind of covariant derivative. Using the non-symmetric metric tensor we find necessary and sufficient conditions for a geodesic mapping $f:\mathbb{GR}_N\to \mathbb{G}\underset 1 {\mathbb{\bar{K}}}{}_N$ with respect to the four kinds of covariant derivatives.

逆变协变外导数的求逆

Inverting covariant exterior derivative

本文提出一种基于庞加莱引理线性同伦算子的算法，用于在星形区域上求逆变协变外导数的逆，以几何和算法方式求解协变常数方程，并讨论了约束条件与解的正则性。

On the symmetry classes of the first covariant derivatives of tensor fields

We show that the symmetry classes of torsion-free covariant derivatives $\nabla T$ of r-times covariant tensor fields T can be characterized by Littlewood-Richardson products $σ[1]$ where $σ$ is a representation of the symmetric group $S_r$ which is connected with the symmetry class of T. If $σ= [λ]$ is irreducible then $σ[1]$ has a multiplicity free reduction $[λ][1] = \sum [μ]$ and all primitive idempotents belonging to that sum can be calculated from a generating idempotent e of the symmetry class of T by means of the irreducible characters or of a discrete Fourier transform of $S_{r+1}$. W…

Generators of algebraic covariant derivative curvature tensors and Young symmetrizers

We show that the space of algebraic covariant derivative curvature tensors R' is generated by Young symmetrized tensor products W*U or U*W, where W and U are covariant tensors of order 2 and 3 whose symmetry classes are irreducible and characterized by the following pairs of partitions: {(2),(3)}, {(2),(2 1)} or {(1 1),(2 1)}. Each of the partitions (2), (3) and (1 1) describes exactly one symmetry class, whereas the partition (2 1) characterizes an infinite set S of irreducible symmetry classes. This set S contains exactly one symmetry class S_0 whose elements U can not play the role of gener…

爱因斯坦-杨-米尔斯场的能量估计及其应用

Energy estimates for the Einstein-Yang-Mills fields and applications

本文为张量非线性波动方程建立外部能量估计，并应用于证明闵可夫斯基时空在爱因斯坦-杨-米尔斯系统下的外部稳定性，为后续$(1+3)$维情形奠定基础。

Higher Affine Connections

For a smooth manifold $M$, it was shown in \cite{BPH} that every affine connection on the tangent bundle $TM$ naturally gives rise to covariant differentiation of multivector fields (MVFs) and differential forms along MVFs. In this paper, we generalize the covariant derivative of \cite{BPH} and construct covariant derivatives along MVFs which are not induced by affine connections on $TM$. We call this more general class of covariant derivatives \textit{higher affine connections}. In addition, we also propose a framework which gives rise to non-induced higher connections; this framework is obta…

Fractional Differential Forms II

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…

The Homotopy Class of twisted $L_\infty$-morphisms

The global formality of Dolgushev depends on the choice of a torsion-free covariant derivative. We prove that the globalized formalities with respect to two different covariant derivatives are homotopic. More explicitly, we derive the statement by proving a more general homotopy equivalence between $L_\infty$-morphisms that are twisted with gauge equivalent Maurer-Cartan elements.