On product affine hyperspheres in ℝn+1

Abstract

In this paper, we study locally strongly convex affine hyperspheres in the unimodular affine space R n+1 which, as Riemannian manifolds, are locally isometric to the Riemannian product of two Riemannian manifolds both possessing constant sectional curvatures. As the main result, a complete classification of such affine hyperspheres is established. Moreover, as direct consequences, affine hyperspheres of dimensions 3 and 4 with parallel Ricci tensor are also classified.

Affine hypersurfaces with constant sectional curvature

“…The method used in the proof of Lemma 4.1 is called the Tsinghua principle. This remarkable technique has been applied in many different situations since its first successful attempt in [1], see [5,6,13,14,21,22] for details.…”

“…By using such approach, some canonical submanifolds with constant sectional curvature (even under more general conditions) in some canonical Riemannian manifold have been classified, cf. [1,5,6,14,21,22], etc. The main purpose of this paper is to classify the hypersurfaces of S 2 × S 2 with constant sectional curvature.…”

Non-existence of conformally flat real hypersurfaces in both the complex quadric and the complex hyperbolic quadric

“…1 The method used in the proof of Lemma 3.1 is elementary and is called the Tsinghua principle. This remarkable technique has been applied in many different situations since its first successful attempt in [8], see [1,6,7,16,29] for details. Essentially, it establishes a bridge between the Codazzi equation and the Ricci identity by calculating the cyclic sum of the second covariant derivative of the second fundamental form.…”

On Product Minimal Lagrangian Submanifolds in Complex Space Forms

“…Let us describe first a general method for choosing suitable orthonormal vectors at a point on M n , which will be used recurrently in the proof of Lemma 3.5. The main idea originates from the very similar situation in studying affine hyperspheres in [9,13,22].…”

“…The second step is then expressing the second fundamental form of the submanifold M n with respect to a conveniently chosen frame. To do so, we proceed by induction (see [22] and [9]). One should notice that, eventually, our main result follows directly from the theorems in [17].…”