Related papers: The geometry of generalised Cheeger-Gromoll metric…

We construct finite element approximations of the Levi-Civita connection and its curvature on triangulations of oriented two-dimensional manifolds. Our construction relies on the Regge finite elements, which are piecewise polynomial…

We investigate horizontal conformality of a differential of a map between Riemannian manifolds where the tangent bundles are equipped with Cheeger--Gromoll type metrics. As a corollary, we characterize the differential of a map as a…

We investigate the structure of a Finsler manifold of nonnegative weighted Ricci curvature including a straight line, and extend the classical Cheeger-Gromoll-Lichnerowicz splitting theorem. Such a space admits a diffeomorphic,…

We formulate quantum group Riemannian geometry as a gauge theory of quantum differential forms. We first develop (and slightly generalise) classical Riemannian geometry in a self-dual manner as a principal bundle frame resolution and a dual…

We prove that the universal Teichmuller space T(1) carries a new structure of a complex Hilbert manifold. We show that the connected component of the identity of T(1), the Hilbert submanifold T_{0}(1), is a topological group. We define a…

Let (M,g) be a pseudo-Riemannian manifold and $T^2M$ be its the second-order tangent bundle equipped with the deformed 2-nd lift metric g which obtained from the 2-nd lift metric by deforming the horizontal part with a symmetric…

Let $(M,g)$ be an $n-$dimensional Riemannian manifold and $T_{1}^{1}(M)$ be its $(1,1)-$tensor bundle equipped with the rescaled Sasaki type metric $% ^{S}g_{f}$ which rescale the horizontal part by a nonzero differentiable function $f$. In…

Here (the last paper in a series of four) we end our presentation of the basics of a systematical approach to the differential geometry of a smooth manifold M (supporting a metric field g and a general connection del) which uses the…

We show Riemannian geometry could be studied by identifying the tangent bundle of a Riemannian manifold $\mathcal{M}$ with a subbundle of the trivial bundle $\mathcal{M} \times \mathcal{E}$, obtained by embedding $\mathcal{M}$…

This is a survey of our recent results on the geometry of moduli spaces and Teichmuller spaces of Riemann surfaces appeared in math.DG/0403068 and math.DG/0409220. We introduce new metrics on the moduli and the Teichmuller spaces of Riemann…

The metric tensor of a Riemannian manifold can be approximated using Regge finite elements and such approximations can be used to compute approximations to the Gauss curvature and the Levi-Civita connection of the manifold. It is shown that…

Employing a class of generalized connections, we describe certain differential complices $\left(\tilde \Omega^*_{\mathbb{T}}(M), \tilde{\mathbb{d}}^{\mathbb{T}}\right)$ constructed from $\wedge^* \mathbb{T} M$ and study some of their basic…

Natural metric structures on the tangent bundle and tangent sphere bundles $S_rM$ of a Riemannian manifold $M$ with radius function $r$ enclose many important unsolved problems. Admitting metric connections on $M$ with torsion, we deduce…

We prove that Riemannian metrics with an absolute Ricci curvature bound and a conjugate radius bound can be smoothed to having a sectional curvature bound. Using this we derive a number of results about structures of manifolds with Ricci…

In this note, we derive explicit formulae for the curvature of a convex sum of Riemannian metrics, \(g_t = (1-t)g_0 + t g_1\). We study whether such a deformation can increase the \emph{average} of the Riemann curvature component…

Sets related to positively curved invariant Riemannian metrics on generalized Wallach spaces are considered. The problem arises in studying of the evolution of such metrics under the normalized Ricci flow equation. For Riemannian metrics of…

The difference tensor R.C-C.R of a semi-Riemannian manifold (M,g), dim M > 3, formed by its Riemannian-Christoffel curvature tensor R and the Weyl conformal curvature tensor C, under some assumptions, can be expressed as a linear…

The L\'evi-Civita connection of a Riemannian manifold is a metric (compatible) linear connection, uniquely determined by its vanishing torsion. It is extremal in the sense that it has minimal torsion at each point. We can extend this idea…

Using Gauss's square-roots of the metric components, the diagonal Riemann tensor components for diagonal metrics are calculated. The result is a form which makes their source in the metric directly intuitive and displays an intriguing…

We define the Ricci curvature, as a measure, for certain singular torsion-free connections on the tangent bundle of a manifold. The definition uses an integral formula and vector-valued half-densities. We give relevant examples in which the…