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In a recent paper the first author introduced two sequences of Riemannian invariants on a Riemannian manifold M, denoted respectively by 6(nl,... ,nk) and 5(ni,... ,nk), which trivially satisfy 6(ni,... ,nk) > 6(n,... , nk). In this article, we completely determine the Riemannian manifolds satisfying the condition 6(nfl,... , nk) = S(nl,... ,nk). By applying the notions of these 6-invariants, we establish new characterizations of Einstein and conformally flat spaces; thus generalizing two well-known results of Singer-Thorpe and of Kulkarni.

For (positive) definite surfaces in R 4 \mathbb {R}^{4} there is a canonical choice of a centroaffine normal plane bundle, which induces a centroaffine invariant Ricci-symmetric connection ∇ \nabla . We classify all surfaces in R 4 \mathbb {R}^{4} with planar ∇ \nabla -geodesics. It turns out that the resulting class of surfaces is umbilical with projectively flat induced connection and flat normal plane bundle.

In this paper we study the affine Gauss maps of nondegenerate surfaces in R 4 {\mathbb {R}^4} with respect to the Burstin-Mayer normalization (1927) and with respect to the normalization obtained by Nomizu and the first author in 1993. We determine up to affine transformations all surfaces in R 4 {\mathbb {R}^4} with constant affine Gauss maps.

In this paper we study the affine Gauss maps of nondegenerate surfaces in R4 with respect to the Burstin-Mayer normalization (1927) and with respect to the normalization obtained by Nomizu and the first author in 1993. We determine up to affine transformations all surfaces in R4 with constant affine Gauss maps.

Let M M be a compact 3 3 -dimensional totally real submanifold of the nearly Kaehler 6 6 -dimensional unit sphere. Let K K be the sectional curvature function of M M . Then, if K > 1 / 16 K > 1/16 , M M is a totally geodesic submanifold (and K ≡ 1 K \equiv 1 ).