Convex mappings between Riemannian manifolds (Q2834822)

In this paper, the authors define the notion of convexity for smooth mappings between Riemannian manifolds in a natural way from classical one, and give some analytical and geometric descriptions.NEWLINENEWLINEA smooth map \(\varphi : (M, g) \to (N, h)\) is called convex(subharmonic) if the corresponding Hessian(Laplacian) \(\nabla g \varphi \geq 0\) ( \(\text{Tr}_g\nabla d \varphi \geq 0\)). One of main result in this paper is the pull-back transport property. Namely, they prove that a smooth map \(\varphi : (M, g) \to (N, h)\) is convex if and only if \(\varphi\) pulls back locally convex partial increasing functions on \(N\) into locally convex functions on \(M\). A smooth function \(f: N^m \to {\mathbb R}\) is called partially locally increasing if there exists an open subset \(V \subset N\) such that the directional derivative \(f_i(q) \geq 0\) for all \(q \in V\) and \(i = 1, \cdots, m\). As a corollary, we can derive thatNEWLINENEWLINERelated to minimal submanifolds and totally geodesic submanifolds, the authors prove that any convex isometric embedding carries totally geodesic submanifolds into convex submanifolds.