Abstract We study Hessian estimators for functions defined over an $n$-dimensional complete analytic Riemannian manifold. introduce new stochastic zeroth-order using $O (1)$ function evaluations. show that, real-valued $f$, our estimator achieves a bias bound of order $ O ( \gamma \delta ^2 ) $, where depends on both the Levi–Civita connection and $\delta is finite difference step size. To best...

We construct a Connes spectral triple or `Dirac operator' on the non-reduced fuzzy sphere $C_\lambda[S^2]$ as realised using quantum Riemannian geometry with central metric $g$ of Euclidean signature and its associated Levi-Civita connection. The Dirac operator is characterised uniquely up to unitary equivalence within our geometric setting an assumption that spinor bundle trivial rank 2 basis....

Given a tame differential calculus over noncommutative algebra [Formula: see text] and an text]-bilinear metric consider the conformal deformation being invertible element of We prove that there exists unique connection on bimodule one-forms which is torsionless compatible with derive concrete formula connecting Levi-Civita for As application, we compute Ricci scalar curvatures general perturba...

We show how an affine connection on a Riemannian manifold occurs naturally as cochain in the complex for Leibniz cohomology of vector fields with coefficients adjoint representation. The coboundary Levi-Civita can be expressed sum two terms, one Laplace-Beltrami operator and other Ricci curvature term. vanishing this has interpretation terms eigenfunctions Laplacian. Separately, we compute cert...

It is shown that for a wide class of analytic Lagrangians which depend only on the scalar curvature of a metric and a connection, the application of the so–called “Palatini formalism”, i.e., treating the metric and the connection as independent variables, leads to “universal” equations. If the dimension n of space–time is greater than two these universal equations are Einstein equations for a g...

Analogous to the characterisation of Brownian motion on a Riemannian manifold as development Euclidean space, we construct sub-Riemannian diffusions equinilpotentisable manifolds by developing canonical stochastic process arising lift an associated model space. The notion introduce for uses Cartan connections, which take place Levi-Civita connection in geometry. We first derive general expressi...

A geometry with parallel skew-symmetric torsion is a Riemannian manifold carrying metric connection torsion. Besides the trivial case of Levi-Civita connection, geometries non-vanishing arise naturally in several geometric contexts, e.g. on reductive homogeneous spaces, nearly Kähler or G2-manifolds, Sasakian and 3-Sasakian manifolds, twistor spaces over quaternion-Kähler manifolds positive sca...

We define a Lévy process on smooth manifold M with connection as projection of solution Marcus stochastic differential equation holonomy bundle M, driven by holonomy-invariant Euclidean space. On Riemannian manifold, our definition (with Levi-Civita connection) generalizes the Eells-Elworthy-Malliavin construction Brownian motion and extends class isotropic introduced in Applebaum Estrade [3]. ...

We consider the metric-affine formulation of bumblebee gravity, derive field equations, and show that connection can be written as Levi-Civita a disformally related metric in which determines disformal part. As consequence, gets coupled to all other matter fields present theory, potentially leading nontrivial phenomenological effects. To explore this issue we compute weak-field, post-Minkowskia...

We consider the sum of Einstein-Hilbert action and a Pontryagin density (PD) in arbitrary even dimension $D$. All curvatures are functions independent affine (torsionless) connections only. In dimension, not only $D=4n$, these first order PD terms shown to be covariant divergences "Chern-Simons" currents. The field equation for connection leads it being Levi-Civita, metric equations equivalent ...