Conformal Riemannian maps between Riemannian manifolds, their harmonicity and decomposition theorems
Let \((M_{1}^{m}, g_{1})\) and \((M_{2}^{n}, g_{2})\) be Riemannian manifolds and \(f: (M_{1}^{m}, g_{1})\longrightarrow (M_{2}^{m}, g_{2})\) a smooth map between them. Then \(f\) is said to be a conformal Riemannian map at \(p_1\in M_1\) if \(0<\text{rank} F_{*p_1}\leq \min\{m, n\}\) and \( F_{*p_1}\) maps the horizontal space \(\mathcal{H}(p_1) = (\ker (F_{*p_1}))\) conformally onto range\((F_{*p_1})\), i.e., there exits a number \(\lambda^{2}(p_1)\neq 0\) such that \[ g_{2}(F_{*p_1}X, F_{*p_1}Y) = \lambda^{2}(p_1) g_{1}(X, Y), \] for \(X\), \(Y\in\mathcal{H}(p_1)\). The author shows that if \(f\) is a surjective homothetic conformal Riemannian map between Riemannian manifolds \((M_{1}^{m}, g_{1})\) and \((M_{2}^{n}, g_{2})\), \(M_1\) is connected and \((M_{1}^{m}, g_{1})\) is complete, then \(f\) is a projection of a locally trivial bundle \(f\) whose base space is also a complete Riemannian manifold. If \(f\) is a proper conformal Riemannian map between Riemannian manifolds \((M_{1}^{m}, g_{1})\) and \((M_{2}^{n}, g_{2})\), \(U\) is a neighborhood of \(p_1\in M_1\), \(P\) a manifold and \(S\) a submersion so that \(F|_{U} = F_{U}= I\circ S\), if \(g_{1U}\) denotes the restriction of \(g_1\) to \(U\) and \(g_P= I^{*}g_2\), then \((U, g_{1U})\) and \((P, g_P)\) are Riemannian manifolds, the submersion \(S: (U, g_{1U})\longrightarrow (P, g_P)\) is a horizontal proper conformal submersion and the immersion \(I: (P, g_P)\longrightarrow (M_{2}^{n}, g_{2})\) is an isometric immersion.
scientific article; zbMATH DE number 5712194
scientific article