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We extend our previous classification of superpotentials of ``scalar curvature type for the cohomogeneity one Ricci-flat equations. We now consider the case not covered in our previous paper, i.e., when some weight vector of the superpotential lies outside (a scaled translate of) the convex hull of the weight vectors associated with the scalar curvature function of the principal orbit. In this situation we show that either the isotropy representation has at most 3 irreducible summands or the first order subsystem associated to the superpotential is of the same form as the Calabi-Yau condition for submersion type metrics on complex line bundles over a Fano Kahler-Einstein product.
We give an overview of our earlier classification results in [DW4] and [DW6] for superpotentials of scalar curvature type of the cohomogeneity one Ricci-flat equations. We then give an account of the classification in the case where the isotropy repr
Classification of curves up to affine transformation in a finite dimensional space was studied by some different methods. In this paper, we achieve the exact formulas of affine invariants via the equivalence problem and in the view of Cartans lemma a
A Ricci soliton $(M,g,v,lambda)$ on a Riemannian manifold $(M,g)$ is said to have concurrent potential field if its potential field $v$ is a concurrent vector field. Ricci solitons arisen from concurrent vector fields on Riemannian manifolds were stu
The main purpose of this article is to provide an alternate proof to a result of Perelman on gradient shrinking solitons. In dimension three we also generalize the result by removing the $kappa$-non-collapsing assumption. In high dimension this new m
The purpose of this work is to close the local deformation problem of rank two Euclidean submanifolds in codimension two by describing their moduli space of deformations. In the process, we provide an explicit simple representation of these submanifo