Conformal immersions of complete Riemannian manifolds and extensions of the Schwarz lemma
A conformal immersion \(\varphi: M\to N\) of a complete connected non-compact Riemannian manifold \((M,g)\) \((m=\dim M\geq 3)\) into another Riemannian manifold \((N,h)\) is weakly distance decreasing (w.d.d.) when \(\varphi^* h= u^{4/ (m-2)} g\) and \(0< u(x) \leq 1\) for all \(x\in M\). The authors provide some conditions on the curvatures of \(M\) and \(N\) which ensure that any conformal immersion \(\varphi\) such that \(s_ N (\varphi (x))\leq s_ M(x)\) \((x\in M)\) is w.d.d. Here, \(s_ M\) and \(s_ N\) are the scalar curvatures of \(M\) and \(N\), respectively.
scientific article; zbMATH DE number 624878
scientific article