Abstract
In the present paper we give a rough classification of exterior differential forms on a Riemannian manifold. We define conformal Killing, closed conformal Killing, coclosed conformal Killing and harmonic forms due to this classification and consider these forms on a Riemannian globally symmetric space and, in particular, on a rank-one Riemannian symmetric space. We prove vanishing theorems for conformal Killing L 2-forms on a Riemannian globally symmetric space of noncompact type. Namely, we prove that every closed or co-closed conformal Killing L 2-form is a parallel form on an arbitrary such manifold. If the volume of it is infinite, then every closed or co-closed conformal Killing L 2-form is identically zero. In addition, we prove vanishing theorems for harmonic forms on some Riemannian globally symmetric spaces of compact type. Namely, we prove that all harmonic one-formsvanish everywhere and every harmonic r -form r 2 is parallel on an arbitrary such manifold. Our proofs are based on the Bochnertechnique and its generalized version that are most elegant and important analytical methods in differential geometry “in the large”.
Highlights
In the present paper we consider conformal Killing, closed conformal Killing, coclosed conformal Killing and harmonic forms which are defined on Riemannian globally symmetric spaces
Our proofs are based on the Bochner technique and its generalized version that are most elegant and important analytical methods in differential geometry “in the large”
The results of the present paper were announced at the International Conference "Differential Geometry" organized by the Banach Center from June 18 to June 24, 2017 at Będlewo (Poland) and at the International Conference "Modern Geometry and its Applications" dedicated to the 225th anniversary of the birth of N.I
Summary
In the present paper we consider conformal Killing, closed conformal Killing, coclosed conformal Killing and harmonic forms which are defined on Riemannian globally symmetric spaces. We prove vanishing theorems for conformal Killing, closed conformal Killing and coclosed conformal Killing L2-forms on Riemannian globally symmetric spaces of noncompact type. We prove vanishing theorems for harmonic forms on some Riemannian globally symmetric spaces of compact type. If M,g is a Riemannian globally symmetric spaces of compact type M,g is a compact Riemannian manifold with non-negative sectional curvature and positive-definite Ricci tensor We can conclude that a -connected and irreducible Riemannian locally symmetric space with positive-definite curvature operator is a Euclidian sphere. M,g is a complete non-compact Riemannian manifold with non-positive sectional curvature and negative-definite Ricci tensor, and diffeomorphic to a Euclidean space M,g is a Riemannian globally symmetric space of noncompact type
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