Related papers: Remarks on Bianchi sums and Pontrjagin classes

We translate into the double forms formalism the basic identities of Greub and Greub-Vanstone that were obtained in the mixed exterior algebra. In particular, we introduce a second product in the space of double forms, namely the…

Let $(X,g)$ be a compact $n$-dimensional smooth Riemannian manifold with a lower bound on the average of the lowest $n-p$ eigenvalues of the curvature operator and the diameter of $X$ is bounded above by $D>0$. In this article, we…

We compute explicit transgression forms for the Euler and Pontrjagin classes of a Riemannian manifold $M$ of dimension 4 under a conformal change of the metric, or a change to a Riemannian connection with torsion. These formulae describe…

In this short note, we provide a quantitative global Poincar\'e inequality for one forms on a closed Riemannian four manifold, in terms of an upper bound on the diameter, a positive lower bound on the volume, and a two-sided bound on Ricci…

Given a complete Riemannian manifold satisfying a weighted Poincar\'{e} inequality and having a bounded below Ricci curvature, various vanishing theorems for harmonic functions and harmonic 1-forms have been published. We generalized these…

The motivation of this paper is to study a second order elliptic operator which appears naturally in Riemannian geometry, for instance in the study of hypersurfaces with constant $r$-mean curvature. We prove a generalized Bochner-type…

A $(p,q)$-double form on a Riemannian manifold $(M,g)$ can be considered simultaneously as a vector-valued differential $p$-form over $M$ or alternatively as a vector-valued $q$-form. Accordingly, the usual Hodge-de Rham Laplacian on…

This paper contains some vanishing theorems for $L^2$ harmonic forms on complete Riemannian manifolds with a weighted Poincar\'e inequality and a certain lower bound of the curvature. The results are in the spirit of Li-Wang and Lam, but…

We prove a new general Poincar\'e-type inequality for differential forms on compact Riemannian manifolds with nonempty boundary. When the boundary is isometrically immersed in Euclidean space, we derive a new inequality involving mean and…

We introduce a natural extension of the metric tensor and the Hodge star operator to the algebra of double forms to study some aspects of the structure of this algebra. These properties are then used to study new Riemannian curvature…

We present a generalization of the topological inequality of Thorpe between the Euler characteristic and $k^{th}$-Pontryagin number of a $4k$-manifold. We also correct and complete some of the arguments from the work of Thorpe in which this…

In this paper, we show several vanishing type theorems for $p$-harmonic $\ell$-forms on Riemannian manifolds ($p\geq2$). First of all, we consider complete non-compact immersed submanifolds $M^n$ of ${N}^{n+m}$ with flat normal bundle, we…

The aim of this article is to introduce a bivariate extension of Shurer-Stancu operators based on (p q)integers. We prove uniform approximation by means of Bohman Korovkin type theorem rate of convergence using total modulus of smoothness…

This paper demonstrates the power of the calculus developed in the two previous parts of the series for all real forms of the almost Hermitian symmetric structures on smooth manifolds, including e.g. conformal Riemannian and almost…

We prove a Poincar\'e, and a general Sobolev type inequalities for functions with compact support defined on a $k$-rectifiable varifold $V$ defined on a complete Riemannian manifold with positive injectivity radius and sectional curvature…

We study critical metrics of higher-order curvature functionals on compact Riemannian $n$-manifolds $(M,g)$. For an integer $k$ with $2 \leq 2k \leq n$, let $R^k$ denote the $k$-th exterior power of the Riemann curvature tensor. We…

We discuss a peculiar interplay between the representation theory of the holonomy group of a Riemannian manifold, the Weitzenboeck formula for the Hodge-Laplace operator on forms and the Lichnerowicz formula for twisted Dirac operators. For…

Given a smooth positive function $f$ defined on the unit circle satisfying a simple condition, we obtain a Poincar\'{e}-type inequality for an arbitrary function $u$ whose weighted average with respect to $f$ is zero. The proof uses…

In this article, under mild constraints on the sectional curvature, we exploit a divergence formula for symmetric endomorphisms to deduce a general Poincar\'e type inequality. We apply such inequality to higher-order mean curvature of…

We prove a vanishing and estimation theorem for the $p^{\text{th}}$-Betti number of closed $n$-dimensional Riemannian manifolds with a lower bound on the average of the lowest $n-p$ eigenvalues of the curvature operator. This generalizes…