This paper deals with the problem of finding pinching conditions for the compactness of a Riemannian manifold. Let \((M,g)\) be a Riemannian manifold, with \(\dim M>2\) and consider the orthogonal decomposition of the Riemannian curvature: \(R={\mathcal W}+U+V\) where \({\mathcal W}\) denotes the Weyl tensor, while \(U\) are \(V\) are the algebraic curvature tensor fields respectively determined by the scalar curvature \(\tau\) and the traceless Ricci tensor \(\rho_0\), that is: \[ U(X,Y,Z,W)=\tfrac\tau{n(n-1)} (g(X,Z)g(Y,W)-g(Y,Z)g (X,W)); \]
\[ V(X,Y,Z,W)=\tfrac 1{n-1}(\rho_0 (X,Z)g(Y,W)-\rho_0(X,W) g(Y,Z)+ \rho_0(Y,W)g (X,Z)-\rho_0 (Y,Z) (Z,W)). \] In [J. Differ. Geom. 21, 47–62 (1985; Zbl 0606.53026)], G. Huisken considered the following pointwise pinching condition: \(\|{\mathcal W}\|^2+ \|V\|^2 \leq\delta_n (1-\varepsilon)^2\|U\|^2\), where \(\varepsilon>0\) and \(\delta_n= {2\over(n-2) (n+1)}\) if \(n\geq 4\), \(n\neq 5\), \(\delta_5= {1\over 10}\). A detailed study of the Ricci flow on complete non compact manifolds with positive scalar curvature, combined with recent results due to R. S. Hamilton, G. Huisken and W. X. Shi, allows to state the following mean theorems.
Theorem 1. Let \((M, g)\) be a complete, \(n\)-dimensional Riemannian manifold with positive and bounded scalar curvature. If \(n\geq 4\) and the pinching condition (1) holds, then \(M\) is compact.
Theorem 2. Let \((M,g)\) be a 3-dimensional, complete, noncompact Riemannian manifold with bounded and non-negative sectional curvatures. If the Ricci tensor \(\rho\) satisfies \(\rho(X,Y) \geq\varepsilon \tau g(X,Y)\), for any vector fields \(X,Y\) and for some \(\varepsilon>0\), then \(M\) is flat.
The paper is endowed with a wide bibliography.