Study of Analysis on Manifolds

(1)证明了一阶自伴随椭圆微分算子的一参数族谱流的分裂公式，该谱流是用超曲面将流形分裂成两个分量时产生的。为了证明这个公式，我们重建了无限维马斯洛夫指数的一般理论，其中应该注意的是，马斯洛夫指数可以通过泛函解析方法定义为任意路径与马斯洛夫循环的交点数。我们还证明了无限维马斯洛夫指数的一个约化定理(2)。我们在四元数投影空间的刺破的共切束上找到了复化的Hopf纤化，并通过两种方法在四元数投影空间上压缩了测地流的两个量化算子。(3)在Cayley射影平原的刺破余切束上构造一个Kahler结构，其Kahler形式与自然辛形式重合。这是通过显式地将其嵌入到8×8复矩阵空间中得到的。(4)证明了3维和4维Heisenberg流形的ζ正则行列式的积分表示，并给出了乘积型罗马尼亚流形的一般公式。然后应用该公式给出高维环面的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.