A GEODESIC EQUATION IN THE SPACE OF SASAKIAN METRICS
A GEODESIC EQUATION IN THE SPACE OF SASAKIAN METRICS
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发表时间：
2009
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通讯作者：
Pengfei Guan;Xi Zhang
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作者：
Pengfei Guan;Xi Zhang
本文讨论了sasaki度量空间上的一个测地线方程。这个方程与sasaki流形的一些有趣的几何性质有关。它可以看作是[11,15,5]中引入的关于Kähler度量空间的著名测地线方程的平行情况。让我们首先回顾一下Kähler度量空间的定义。对于给定的具有Kähler度量g的紧致Kähler流形（M, g），设HK = {φ∈C(M)|（gij′+ φij′）> 0}为g的上同类中所有Kähler度量的空间。Mabuchi[11]在HK上引入了一个自然黎曼结构。[11]的形式化计算表明HK是一个非紧型的非正弯曲对称空间。Semmes[15]研究了Kähler空间HK上的测地线方程，并将其与齐次复Monge-Ampere方程的Dirichlet问题联系起来
This paper is to draw attention to a geodesic equation on space of Sasakian metrics. The equation is connected to some interesting geometric properties of Sasakian manifolds. It can be viewed as a parallel case of a well-known geodesic equation for the space of Kähler metrics introduced in [11, 15, 5]. Let us first recall the definition of the space of Kähler metrics. For a given compact Kähler manifold (M, g) with Kähler metric g. Let HK = {φ ∈ C(M)|(gij̄ + φij̄) > 0} be the space of all Kähler metrics in cohomology class of g. Mabuchi [11] introduced a natural Riemannian structure on HK . A formal calculation in [11] yields that HK is a non-positive curved symmetric space of non-compact type. Semmes [15] studied the geodesic equation on the Kähler space HK relating it to a Dirichlet problem of the homogenous complex Monge-Ampere equation