齐性空间中曲线的平方根速度框架

Real square roots of matrices: differential properties in semi-simple, symmetric and orthogonal cases

We study the differential and metric structures of the set of real square roots of a non-singular real matrix, under the assumption that the matrix and its square roots are semi-simple, or symmetric, or orthogonal.

Two-root Riemannian Manifolds

Osserman manifolds are a generalization of locally two-point homogeneous spaces. We introduce $k$-root manifolds in which the reduced Jacobi operator has exactly $k$ eigenvalues. We investigate one-root and two-root manifolds as another generalization of locally two-point homogeneous spaces. We prove that there is no two-root Riemannian manifold of odd dimension. In twice an odd dimension, we describe all two-root Riemannian algebraic curvature tensors and give additional conditions for two-root Riemannian manifolds.

Applications of square roots of diffeomorphisms

In this paper, we prove that on any contact manifold, there exists an arbitrary C^{\infty}-small contactomorphism which does not admit a square root. In particular, there exists an arbitrary C^{\infty}-small contactomorphism which is not "autonomous". This result is the first step to study the topology of non-autonomous contactomorphisms. As an application, we also prove a similar result for the diffeomorphism group for any smooth manifold.

Rigidity of smooth Schubert varieties in a rational homogeneous manifold associated to a short root

We classify smooth Schubert varieties S_0 in a rational homogeneous manifold S associated to a short root, and show that they are rigid in the sense that any subvariety of S having the same homology class as S_0 is induced by the action of Aut_0(S), unless S_0 is linear.

Anisotropic flow of convex hypersurfaces by the square root of the scalar curvature

We show the existence of a smooth solution for the flow deformed by the square root of the scalar curvature multiplied by a positive anisotropic factor $ψ$ given a strictly convex initial hypersurface in Euclidean space suitably pinched. We also prove the convergence of rescaled surfaces to a smooth limit manifold which is a round sphere. In dimension two, it is shown that, with a volume preserving rescaling, the limit profile satisfies a soliton equation.

平方根速度框架中的最优重参数化

Optimal reparametrizations in the square root velocity framework

平方根速度框架是形状分析中定义曲线和函数数据之间的距离的方法。识别两条曲线，如果它们通过重新参数化而不同，则会导致非参数化曲线的商空间。在本文中，我们研究分析和拓扑方面的建设，这类绝对连续曲线。我们证明了平方根速度变换是一个同胚，并且重新参数化半群的作用是连续的。我们还表明，给定两个$C^1$ -曲线，存在最佳的reparametrisations实现由它们所代表的非参数化曲线之间的最小距离。此外，我们给出了两个Lipschitz曲线的例子，其中不存在最佳重新参数化对。