Theorems of existence and of vanishing of conformally killing forms

Abstract

On an n-dimensional compact, orientable, connected Riemannian manifold, we consider the curvature operator acting on the space of covariant traceless symmetric 2-tensors. We prove that, if the curvature operator is negative, then the manifold admits no nonzero conformally Killing p-forms for p = 1, 2, . . . , n − 1. On the other hand, we prove that the dimension of the vector space of conformally Killing p-forms on an n-dimensional compact simply-connected conformally flat Riemannian manifold (M, g) is not zero.

On the Betti and Tachibana Numbers of Compact Einstein Manifolds

“…The condition is an analog of the well known Poincaré duality for Betti numbers. Moreover, we proved in [18,19] that Tachibana numbers t 1 (M ), . .…”

“…The vector space of conformal Killing p-forms on a compact Riemannian manifold (M, g) has finite dimension t p (M ) named the Tachibana number (see e.g. [17,18,19]). Tachibana numbers t 1 (M ), .…”

On higher-order Codazzi tensors on complete Riemannian manifolds

“…where ∆ B = div • grad is the Laplace-Beltrami operator, In the next paragraph we will prove that Q P ( It is famous for its numerous applications [2, pp. 51-52; 346-347]; [31]; [32]; [34]; [35].…”

The Sampson Laplacian on Negatively Pinched Riemannian Manifolds

“…The vector space of conformal Killing p-forms on an n-dimensional closed Riemannian manifold (M, g) has a finite dimension t p (M ) named the Tachibana number (e.g., [22,29,35]). The numbers t 1 (M ), .…”