An Integral Formula on the Scalar Curvature of Algebraic Manifolds

Abstract

Abstract. It is proved in this note that if the scalar curvature of an n-dimensional algebraic complex submanifold is bigger than n2, then it is totally geodesic.

On Compact Kaehler Submanifolds in ℂP n + p with Nonnegative Sectional Curvature

“…The hypothesis of the lemma implies that f(v) > 0. Thus, the inequality follows directly from(5). Furthermore, by(4), we see that the equality holds if and only ifH(u)>l-2\\a(v,v)\\2= \ for all u e UM.…”

“…There are a number of conjectures for Kaehler submanifolds M" in CPn+p suggested by K. Ogiue ([1]); some have been resolved under a suitable topological restriction (e.g., Mn is complete) (cf. [2,3,4,5]). In this direction, one of the open problems so far is as follows.…”

Riemannian Submanifolds: A Survey

“…J. H. Cheng (1981) and R. J. Liao (1988) characterized complex quadric hypersurface in terms of its scalar curvature:…”

The rigidity on the second fundamental form of projective manifolds

“…Ogiue also posed several open problems in [Og74,p. 112] to characterize the totally geodesic submanifolds in P n+r in terms of various curvature pinching conditions, and some of them have been answered or partially answered ( [Ch81], [Ro85], [RV84], [Li87], [Li88]), to the author's best knowledge. In particular, Ogiue conjectured that ([Og74, p. 112, Problem 3]) if |σ| 2 < n everywhere on a complete complex n-dimensional holomorphically immersed submanifold M in P n+r , then it must be totally geodesic.…”

“…The key idea in Simons's paper [Si68] is to calculate the Laplacian of |σ| 2 and estimate some terms involved to produce the desired results. This idea was more or less inherited by most papers related to the second fundamental form of complex submanifolds in complex projective spaces (e.g., those summarized in [Og74]), but with one exception in [Ch81], where some arguments are of algeo-geometric nature. This motivates us to apply deeper results in algebraic geometry to treat this kind of problems, and the main tools employed in our paper are some classification results in the adjunction theory of algebraic geometry, mainly due to Fujita ([Fu90]).…”