Study of Analysis on Manifolds

(1)证明了一类一阶自伴椭圆微分算子族的谱流的分裂公式，它是流形被超曲面分裂成两个分量时产生的。为了证明这个公式，我们重新构造了无穷维Maslov指标的一般理论，其中值得注意的是，对于任意路径，利用泛函分析方法可以将Maslov指标定义为与Maslov圈的交数。我们还证明了无限维Maslov指标的一个约化定理。(2)我们发现了四元数射影空间的穿孔余角丛上的复杂化的Hopf纤维，并用两种方法压缩了四元数射影空间上测地流的两个量化算子。(3)在Cayley射影平面的穿孔余切丛上构造了Kahler结构，它的Kahler形式与自然辛形式重合。这是通过显式嵌入到8×8复矩阵空间中得到的。(4)证明了三维和四维Heisenberg流形的Zeta正则化行列式的积分表示，并给出了积型罗马尼亚流形的一般公式。然后利用该公式给出了高维环面的Zeta正则化行列式的表达式。

(1)We proved a splitting formula for a spectral flow of a one parameter family of first order selfadjoint elliptic differential operators which arises when we split a manifold into two components by a hypersurface.To prove this formula we reconstruct a general theory of infinite dimensional Maslov index, where it should be noted that the Maslov index can be defined for arbitrary paths as an intersection number with a Maslov cycle by means of a functional 'analytic method.We also proved a reduction theorem of the Maslov index in the infinite dimension(2)We found the complexified Hopf fiberation on the punctured cotangent bundle of the quaternion projective space, and constrict two quantization operators of the geodesic flow on quaternion projective spaces by two methods.(3)We construct a Kahler structure on the punctured cotangent bundle of the Cayley projective plain whose Kahler form coincides with the natural symplectic form.This is given by explicitly embedding it into the space of 8×8 complex matrices.(4)We proved an integral representation of the zeta-regularized determinant of 3 and 4 dimensional Heisenberg manifolds, and also give a general formula for product type Romanian manifolds.Then we apply the formula to give an expression for the zeta-regularized determinant of the higher dimensional torus.