Estimates on the non-compact expanding gradient Ricci solitons

In this paper, we deal with the complete non-compact expanding gradient Ricci soliton (Mn,g) with positive Ricci curvature. On the condition that the Ricci curvature is positive and the scalar curvature approaches 0 towards infinity, we derive a useful estimate on the growth of potential functions. Based on this and under the same assumptions, we prove that ∫t0 Rc (γ'(s) , γ' (s))ds and ∫t0 Rc (γ' (,s). v)ds at least have linear growth, where 7(5) is a minimal normal geodesic emanating from the point where R obtains its maximum. Furthermore, some other results on the Ricci curvature are also obtained.

On Cheng's eigenvalue comparison theorem

We observe that Cheng's Eigenvalue Comparison Theorem for normal geodesic balls [4] is still valid if we impose bounds on the mean curvature of the distance spheres instead of bounds on the sectional and Ricci curvatures. In this version, there is a weak form of rigidity in case of equality of the eigenvalues. Namely, equality of the eigenvalues implies that the distance spheres of the same radius on each ball has the same mean curvature. On the other hand, we construct smooth metrics $g_{\kappa}$ on $[0,r]\times \mathbb{S}^{3}$, non-isometric to the standard metric canκ of constant sectional curvature κ, such that the geodesic balls $B_{g_{\kappa}}(r)=([0,r]\times \mathbb{S}^{3},g_{\kappa})$, $B_{{\rm can}_{\kappa}}(r)=([0,r]\times \mathbb{S}^{3},{\rm can}_{\kappa})$ have the same first eigenvalue, the same volume and the distance spheres $\partial B_{g_{\kappa}}(s)$ and$\partial B_{{\rm can}_{\kappa}}(s)$, $0<s\leq r$, have the same mean curvatures. In the end, we apply this version of Cheng's Eigenvalue Comparison Theorem to construct examples of Riemannian manifolds M with arbitrary topology with positive fundamental tone λ*(M)>0 extending Veeravalli's examples,[7]

A characterization of harmonic foliations by the volume preserving property of the normal geodesic flow

We prove that a Riemannian foliation with the flat normal connection on a Riemannian manifold is harmonic if and only if the geodesic flow on the normal bundle preserves the Riemannian volume form of the canonical metric defined by the adapted connection.