We consider surfaces with parallel mean curvature vector (pmc surfaces) in complex space forms and introduce a holomorphic differential on such surfaces. When the complex dimension of the ambient space is equal to two we find a second holomorphic differential and then determine those pmc surfaces on which both differentials vanish. We also provide a reduction of codimension theorem and prove a non-existence result for pmc 2-spheres in complex space forms.

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We prove that the quotient space of a variationally complete group action is a good Riemannian orbifold. The result is generalized to singular Riemannian foliations without horizontal conjugate points.

In this paper we investigate a kind of generalized Ricci flow which possesses a gradient form. We study the monotonicity of the given function under the generalized Ricci flow and prove that the related system of partial differential equations are strictly and uniformly parabolic. Based on this, we show that the generalized Ricci flow defined on an n-dimensional compact Riemannian manifold admits a unique short-time smooth solution. Moreover, we also derive the evolution equations for the curvatures, which play an important role in our future study.

Unit tangent bundle of a surface carries various information of tangent vector fields on that surface. For 2-spheres (ie genus-zero closed surfaces), the unit tangent bundle is a closed 3-manifold that has non-trivial topology and cannot be embedded in R3. Therefore it cannot be constructed by existing mesh generation algorithms directly. This work aims at the first discrete construction of unit tangent bundles over 2-spheres using tetrahedral meshes. We propose a two-stage algorithm for the construction, which starts from constructing two local bundles and then combines them into a global bundle.