Sugato Mukhopadhyay
We prove that the $4D_\pm$ calculi on the quantum group $SU_q(2)$ satisfy a metric-independent sufficient condition for the existence of a unique bicovariant Levi-Civita connection corresponding to every bi-invariant pseudo-Riemannian metric.
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We study covariant derivatives on a class of centered bimodules $\mathcal{E}$ over an algebra A. We begin by identifying a $\mathbb{Z} ( A ) $-submodule $ \mathcal{X} ( A ) $ which can be viewed as the analogue of vector fields in this context; $ \mathcal{X} ( A ) $ is proven to be a Lie algebra. Connections on $\mathcal{E}$ are in one to one correspondence with covariant derivatives on $ \mathcal{X} ( A ). $ We recover the classical formulas of torsion and metric compatibility of a connection in the covariant derivative form. As a result, a Koszul formula for the Levi-Civita connection is als…
On the existence of noncommutative Levi-Civita connections in derivation based calculi
本研究在*代数上基于导子的非交换微分微积分框架下，给出Levi-Civita联络存在的充要条件与推广的度量对称条件，通过秩3自由模的显式计算阐释理论，采用纯代数方法。
We explore the possibility of introducing q-deformed connections on the quantum 2-sphere and 3-sphere, satisfying a twisted Leibniz rule in analogy with q-deformed derivations. We show that such connections always exist on projective modules. Furthermore, a condition for metric compatibility is introduced, and an explicit formula is given, parametrizing all metric connections on a free module. For the module of 1-forms on the quantum 3-sphere, a q-deformed torsion freeness condition is introduced and we derive explicit expressions for the Christoffel symbols of a Levi-Civita connection for a g…
We introduce $q$-deformed connections on the quantum 2-sphere and 3-sphere, satisfying a twisted Leibniz rule in analogy with $q$-deformed derivations. We show that such connections always exist on projective modules. Furthermore, a condition for metric compatibility is introduced, and an explicit formula is given, parametrizing all metric connections on a free module. On the quantum 3-sphere, a q-deformed torsion freeness condition is introduced and we derive explicit expressions for the Christoffel symbols of a Levi-Civita connection for a general class of metrics. We also give metric connec…
Metric compatibility and Levi-Civita Connections on Quantum Groups
研究 Hopf 代数上的任意联络，提出度量相容联络定义，并以 H 值矩阵可逆为判据刻画 Levi-Civita 联络的存在唯一性。
We study connections on hermitian modules, and show that metric connections exist on regular hermitian modules; i.e finitely generated projective modules together with a non-singular hermitian form. In addition, we develop an index calculus for such modules, and provide a characterization in terms of the existence of a pseudo-inverse of the matrix representing the hermitian form with respect to a set of generators. As a first illustration of the above concepts, we find metric connections on the fuzzy sphere. Finally, the framework is applied to a class of noncommutative minimal surfaces, for w…
The Levi-Civita connection and Chern connections for cocycle deformations of Kähler manifolds
本文研究了凯勒流形的共积变形，证明了变形后的陈连接是扭曲全纯和反全纯双模连接的直接和，并探讨了复结构和全纯双模的扭曲性质。
Comparison of Levi-Civita connections in noncommutative geometry
本文比较了三种非交换几何中Levi-Civita联络的构造，提供直接转换，并扩展了Hermitian无挠联络的存在性结果。
Levi-Civita connection on the irreducible quantum flag manifolds
本文对一类量子齐次空间上的协变度量进行分类，证明了在不可约量子旗流形的 Heckenberger--Kolb 微积分上存在唯一的量子对称协变度量，并证明了任意实协变度量的 Levi-Civita 联络存在且唯一，推广了量子射影空间的结果。
We prove a Koszul formula for the Levi-Civita connection for any pseudo-Riemannian bilinear metric on a class of centered bimodule of noncommutative one-forms. As an application to the Koszul formula, we show that our Levi-Civita connection is a bimodule connection. We construct a spectral triple on a fuzzy sphere and compute the scalar curvature for the Levi-Civita connection associated to a canonical metric.
Projective real calculi and Levi-Civita connections
本文研究了射影模上实微积分的Levi-Civita联络存在性，以显式投影系数给出了充分必要条件，并重点分析了复N×N矩阵代数上N维复向量单模的情形。
For bicovariant differential calculi on quantum groups various notions on connections and metrics (bicovariant connections, invariant metrics, the compatibility of a connection with a metric, Levi-Civita connections) are introduced and studied. It is proved that for the bicovariant differential calculi on $SL_{q}(N)$, $O_{q}(N)$ and $Sp_{q}(N)$ from the classification of Schmuedgen, K. and Schueler, A. there exist unique Levi-Civita connections.
Existence and uniqueness of the Levi-Civita connection on noncommutative differential forms
该研究结合希尔伯特模与代数方法，给出非交换微分形式上埃尔米特无挠联络的存在充要条件与唯一充分条件，证明紧黎曼流形θ-形变存在唯一该类联络并显式构造，还得到经典Levi-Civita联络的新构造。
We prove the existence and uniqueness of Levi-Civita connections for strongly sigma-compatible pseudo-Riemannian metrics on tame differential calculi. Such pseudo-Riemannian metrics properly contain the classes of bilinear metrics as well as their conformal deformations. This extends the previous results in references 9 and 10. Star-compatibility of Levi-Civita connections for bilinear pseudo-Riemannian metrics are also discussed.
We construct tame differential calculi coming from toral actions on a class of $\mathrm{C}^*$-algebras. Relying on the existence of a unique Levi-Civita connection on such a calculus, we prove a version of the Bianchi identity. A Gauss-Bonnet theorem for the canonical tame calculus of rank two is studied.
We give a new definition of Levi-Civita connection for a noncommutative pseudo-Riemannian metric on a noncommutative manifold given by a spectral triple. We prove the existence-uniqueness result for a class of modules of one forms over a large class of noncommutative manifolds, including the matrix geometry of the fuzzy 3-sphere, the quantum Heisenberg manifolds and Connes-Landi deformations of spectral triples on the Connes-Dubois Violette-Rieffel-deformation of a compact manifold equipped with a free toral action. It is interesting to note that in the example of the quantum Heisenberg manifo…
Fubini-Study metrics and Levi-Civita connections on quantum projective spaces
本文在量子射影空间上基于Heckenberger-Kolb微分演算构造了Fubini-Study度量的量子类似物及相应Levi-Civita联络，证明其无挠、无余挠且为满足更强度量相容性的双模联络。
Cartan structure equations and Levi-Civita connection in noncommutative geometry
研究带有三角Hopf代数作用且非交换性与相关辫子兼容的代数上的微分和黎曼几何，通过扩展Cartan演算证明了曲率和挠率的向量场与形式方法的等价性，推导了Cartan结构方程和Bianchi恒等式，并利用Koszul公式证明了任意伪黎曼度量的Levi-Civita联络的存在唯一性。
Given a tame differential calculus over a noncommutative algebra $\mathcal{A}$ and an $\mathcal{A}$-bilinear pseudo-Riemannian metric $g_0,$ consider the conformal deformation $ g = k. g_0, $ $k$ being an invertible element of $\mathcal{A}.$We prove that there exists a unique connection $\nabla$ on the bimodule of one-forms of the differential calculus which is torsionless and compatible with $g.$ We derive a concrete formula connecting $\nabla$ and the Levi-Civita connection for the pseudo-Riemannian metric $g_0.$ As an application, we compute the Ricci and scalar curvature for a general conf…
Noncommutative Riemannian geometry of Kronecker algebras
本文研究Kronecker箭图路径代数的非交换黎曼几何，构造无挠双模联络及Levi-Civita联络，并探讨其与厄米形式的相容性。
Two toy models are considered within the framework of noncommutative differential geometry. In the first one, the Einstein action of the Levi-Civita connection is computed for the algebra of matrix valued functions on a torus. It is shown that, assuming some constraints on the metric, this action splits into a classical-like, a quantum-like and a mixed term. In the second model, an analogue of the Palatini method of variation is applied to obtain critical points of the Einstein action functional for $M\sb 4(R)$. It is pointed out that a solution to the Palatini variational problem is not neces…
Geometric Dirac operator on noncommutative torus and $M_2(\Bbb C)$
该研究针对非交换环面与$M_2(\Bbb C)$代数，求解了量子几何实现的Dirac算子与谱三元组，分类了不同几何参数下的谱三元组类型，并阐释了完全几何实现情形下的Lichnerowicz公式。
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