On Isometric and Minimal Isometric Embeddings

In this paper we study critial isometric and minimal isometric embeddings of classes of Riemannian metrics which we call quasi-κ-curved metrics. Quasi-κ-curved metrics generalize the metrics of space forms. We construct explicit examples and prove results about existence and rigidity. Introduction Definition: Let (M, g̃) be a Riemannian manifold. We will say g̃ is a quasi-κcurved metric if there exists a smooth positive definite quadratic form Q on M such that for all x ∈ M (1) Rx = −γ(Qx, Qx) + (κ+ 1)γ(g̃x, g̃x) where γ : ST ∗ → S(ΛT ∗) denotes the algebraic Gauss mapping and Rx the Riemann curvature tensor. (See §1. for more details.) Quasi-κ-curved metrics are a generalization of quasi-hyperbolic metrics defined in [BBG], which correspond to κ = −1. We will also refer to the case κ = 0 as quasi-flat metrics. We will assume that n ≥ 3. When n = 3, the quasi-κ-curved condition is an open condition on the metric, and thus in this case the class of metrics we study is quite general. The condition is stronger in higher dimensions. In this paper we study local isometric embeddings and minimal isometric embeddings of quasi-κ-curved manifolds. Before giving our results, it will be useful to review some of what is known: 1991 Mathematics Subject Classification. 53C42, 53C25, 58A15.