Related papers: Reconstruction and quantization of Riemannian stru…
We consider a regular distribution $\mathcal{D}$ in a Riemannian manifold $(M,g)$. The Levi-Civita connection on $(M,g)$ together with the orthogonal projection allow to endow the space of sections of $\mathcal{D}$ with a natural covariant…
On a Riemannian or a semi-Riemannian manifold, the metric determines invariants like the Levi-Civita connection and the Riemann curvature. If the metric becomes degenerate (as in singular semi-Riemannian geometry), these constructions no…
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…
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…
Noncommutative or quantum Riemannian geometry has been proposed as an effective theory for aspects of quantum gravity. Here the metric is an invertible bimodule map $\Omega^1\otimes_A\Omega^1\to A$ where $A$ is a possibly noncommutative or…
We overview a new mechanism whereby classical Riemannian geometry emerges out of the differential structure on quantum spacetime, as extension data for the classical algebra of differential forms. Outcomes for physics include a new formula…
We develop the formalism for noncommutative differential geometry and Riemmannian geometry to take full account of the *-algebra structure on the (possibly noncommutative) coordinate ring and the bimodule structure on the differential…
We study the differential and Riemannian geometry of algebras $A$ endowed with an action of a triangular Hopf algebra $H$ and noncommutativity compatible with the associated braiding. The modules of one forms and of braided derivations are…
The main purpose of this article is to reconstruct the nonnegative coefficient $a$ in the double phase problem $\mathrm{div}\,(|\nabla u|^{p-2}\nabla u+a|\nabla u|^{q-2}\nabla u)=0$ in a domain $\Omega$, $u=f$ on $\partial\Omega$, from the…
Let $(M,g)$ be a Riemannian manifold, and $m$ be a second metric on $M$. We give expressions of $m$'s associated connection, and Riemann curvature tensor $R_m$, in terms of $R_g$ and certain combinations of covariant derivatives of $m$…
We combine Hilbert module and algebraic techniques to give necessary and sufficient conditions for the existence of an Hermitian torsion-free connection on the bimodule of differential one-forms of a first order differential calculus. In…
We provide an explicit formula for the Levi-Civita connection and Riemannian Hessian for a Riemannian manifold that is a quotient of a manifold embedded in an inner product space with a non-constant metric function. Together with a…
This paper investigates the algebraic and geometric consequences of the associativity of the symmetric part $U$ of the Levi-Civita connection on a pseudo-Riemannian Lie algebra $(\mathfrak{g}, \langle \cdot, \cdot \rangle)$. We demonstrate…
In this paper we consider an inverse coefficients problem for a quasilinear elliptic equation of divergence form $\nabla\cdot\vec{C}(x,\nabla u(x))=0$, in a bounded smooth domain $\Omega$. We assume that…
A differential calculus on Cuntz algebra with three generators coming from the action of rotation group in three dimensions is introduced. The differential calculus is shown to satisfy Assumptions I-IV of [1] so that Levi-Civita Connection…
We study aspects of noncommutative Riemannian geometry of the path algebra arising from the Kronecker quiver with N arrows. To start with, the framework of derivation based differential calculi is recalled together with a discussion on…
Let $M$ be an $n-$dimensional differentiable manifold with a symmetric connection $\nabla $ and $T^{\ast}M$ be its cotangent bundle. In this paper, we study some properties of the modified Riemannian extension $% \widetilde{g}_{\nabla,c}$…
We establish that any affine manifold $(M,\nabla)$ endowed with a parallel volume form $\omega,$ admits, in any conformal class of Riemannian metrics, a representative $H$ for which $\nabla$ is the Levi-Civita connection. This provides a…
We present an extended version of Riemannian geometry suitable for the description of current formulations of double field theory (DFT). This framework is based on graded manifolds and it yields extended notions of symmetries, dynamical…
We introduce a method in differential geometry to study the derivative operators of Siegel modular forms. By determining the coefficients of the invariant Levi-Civita connection on a Siegel upper half plane, and further by calculating the…