S.S. Chern raised the problem to find necessary and sufficient conditions for a given Riemannian manifold to be realizable on a minimal submanifold of a Euclidean space. The aim of this paper is to provide new necessary conditions. For minimal submanifolds in a Euclidean space we consider the negative of the Ricci tensor as defining a new metric, which is nothing but the third fundamental form,...

Let (Mn, g) be a compact Kähler manifold with nonpositive bisectional curvature. We show that a finite cover is biholomorphic and isometric to a flat torus bundle over a compact Kähler manifold Nk with c1 < 0. This confirms a conjecture of Yau. As a corollary, for any compact Kähler manifold with nonpositive bisectional curvature, the Kodaira dimension is equal to the maximal rank of the Ricci ...

We study some properties of a half-lightlike submanifoldM , of a semi-Riemannianmanifold, whose shape operator is conformal to the shape operator of its screen distribution. We show that any screen distribution S(TM) of M is integrable and the geometry of M has a close relation with the nondegenerate geometry of a leaf of S(TM). We prove some results on symmetric induced Ricci tensor and null s...

The Newman-Penrose equations for spacetimes having one spacelike Killing vector are reduced – in a geometrically defined “canonical frame” – to a minimal set, and its differential structure is studied. Expressions for the frame vectors in an arbitrary coordinate basis are given, and coordinate-independent choices of the metric functions are suggested which make the components of the Ricci tenso...

It is the purpose of the present paper to outline an introduction in theory of embeddings in the 2-osculator bundle. First, we recall the notion of 2-osculator bundle ([1],[2]) and the notion of submanifolds in the 2-osculator bundle. A moving frame is constructed. The induced connections and the relative covariant derivation are discussed in third and fourth sections.The Ricci identities for t...

One discusses here the connection between σ-model gauge anomalies and the existence of a connection with torsion that does not flattten the Ricci tensor of the target manifold, by considering a number of non-symmetric coset spaces. The influence of an eventual anisotropy on a certain direction of the target manifold is also contemplated. Pacs numbers: 02.40.-k; 11.10.-z; 11.10.Lm. dfranco@cbpf....

Let G be a homotopically trivial and effective compact Lie group action on a compact manifold N of nonpositive curvature. Under certain assumptions on N we prove that if G has dimension equal to rank of Center π1(N), then G must be connected. Furthermore, if on N there exists a point having negative definite Ricci tensor, then we show that G is the trivial group.

A Riemannian manifold is called harmonic if its volume density function expressed in polar coordinates centered at any point of the manifold is radial. Flat and rank-one symmetric spaces are harmonic. The converse (the Lichnerowicz Conjecture) is true for manifolds of nonnegative scalar curvature and for some other classes of manifolds, but is not true in general: there exists a family of homog...

We describe and construct here pseudo-Hermitian structures θ without torsion (i.e. with transversal symmetry) whose Webster-Ricci curvature tensor is a constant multiple of the exterior differential dθ. We call these structures pseudo-Hermitian Einstein and our result states that they all can be derived locally from Kähler-Einstein metrics. Moreover, we discuss the corresponding Fefferman metri...

In this note we will adapt Topping’s L-optimal transportation theory for Ricci flow to a more general situation, i.e. to a closed manifold (M, gij(t)) evolving by ∂tgij = −2Sij , where Sij is a symmetric tensor field of (2,0)-type on M . We extend some recent results of Topping, Lott and Brendle, generalize the monotonicity of List’s (and hence also of Perelman’s) W-entropy, and recover the mon...