Study of Analysis on Manifolds

我们主要研究了亚黎曼流形的解析和几何方面的问题。我们的子黎曼流形是强意义上的，也就是说，子黎曼结构是由一个非完整子束来定义的，这个子束作为向量束是平凡的。在这样的流形上我们有一个次椭圆算子，我们称之为次拉普拉斯算子。这种流形的例子是三维和七维球体，或零流形，我们从解析延拓，剩余演算和zeta正则行列式的显式确定的角度研究了它们的谱zeta函数。此外，我们还研究了从总空间上的子拉普拉斯算子到基流形的亚椭圆算子的一般框架，并得到了它们的双特征流之间的关系。特别是通过Hopf纤维束S^3→CP^1发现了S^3上的次黎曼测地线与CP^1上曲线的等周方面的关系，并考虑了S^3上由左右四元数乘法结构定义的双纤维，确定了球面Grushin算子的双特征流。

Mainly we focused our study on the topics on analytic and geometric aspects of sub-Riemannian manifolds. Our sub-Riemannian manifolds are in the strong sense, that is、the sub-Riemannian structure is defined by a non-holonomic sub-bundle which is trivial as a vector bundle. So we have a sub-elliptic operator on such manifolds, which we call sub-Laplacian. Examples of such manifolds are three and seven dimensional spheres, or nilmanifolds and we studied their spectral zeta functions from the point of views of analytic continuation, residue calculus and the explicit determination of zeta regularized determinant. Also, we investigate a general framework to define a sub-elliptic operator from a sub-Laplacian on the total space to the base manifold through a submersion (or more specifically fiber bundle setting) and obtained a relation between their bi-characteristic flows. Especially we find a relation between sub-Riemannian geodesics on S^3 through Hopf fiber bundle S^3→CP^1 by notifying their relation with the isoperimetric aspect of curves on CP^1 and by considering double fiberings defined by left and right quaternion multiplication structure on S^3, we determine bi-characteristic flow of the spherical Grushin operator.