On geometric and stochastic mean values for small geodesic spheres in Riemannian manifolds
On geometric and stochastic mean values for small geodesic spheres in Riemannian manifolds
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关于黎曼流形中小测地线球体的几何和随机平均值
DOI：
10.21099/tkbjm/1496160508
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发表时间：
1987
影响因子：
0.7
通讯作者：
Y. Ogura
中科院分区：
文献类型：
--
作者：
M. Kôzaki;Y. Ogura
许多作者最近研究了黎曼流形中小测地球的第一或第二平均值（或它们之间的关系）来表征调和空间、爱因斯坦空间和超爱因斯坦空间（[3]、[7]、[10]、[17]等）。其中，0.Kowalski[10]通过两个平均值在某种意义上的一致程度来表征这三个空间，提出了应位于调和空间和超爱因斯坦空间之间的新类空间。另一方面，M. Pinsky [12] 证实随机平均值对于描述爱因斯坦空间也很有用。在本文中，我们对上述三个平均值进行了更详细的研究，并填补了前人工作中的空白（下面的定理2）。我们证明的主要工具是 Schauder 估计，它使我们能够比之前大多数论文中使用的 Cauchy-Kowalewski 处理 C°流形的方法更容易地处理 C°流形。我们还引入了一些其他新条件，它们也表征了上述三个空间，即条件（M2）、（M4）和（L2）-（L4）（定义见第2节）。条件（M2）是[3]中给出的（M3）的变体。但它与(M1)的联系似乎比(M3)更自然。条件 (M4) 和 (L4) 与 [8: p. 11] 中的 Helgason 展开密切相关。 435]。事实上，如果流形是欧几里得空间或一阶全局对称空间，则上述三个平均值与赫尔加森的平均值一致，并且我们感兴趣的是拉普拉斯算子在多大程度上决定了平均值。我们的结果包括断言；当且仅当流形是调和的时，由三个平均值导出的算子之一可以通过拉普拉斯多项式序列进行展开（定理 2 (1)）。作为副产品，我们还获得了 C6 流形解析的一些充分条件（定理 1）。在我们的证明过程中，我们部分地回答了[10]中给出的Kowalski猜想的光滑流形（定理3）。
The characterization of the harmonic, Einstein and super-Einstein spaces bymeans of the firstor the second mean values (or the relations between them) for small geodesic spheres in a Riemannian manifold is recently studied by many authors ([3], [7], [10], [17], etc.). Among them, 0. Kowalski [10] characterized the three spaces by the degree of concordance of the two mean values in some sense, proposing new classes of spaces which should be located between the harmonic and the super-Einstein spaces. On the other hand M. Pinsky [12] verified that the stochastic mean values are also useful for the characterization of the Einstein spaces. In this paper, we study the above three mean values more in detail and fillthe blanks in the previous works (Theorem 2 below). The main tool for our proof is Schauder's estimate, which enables us to treat C°°manifolds even more easily than Cauchy-Kowalewski's method for C° manifolds used in most of the previous papers. We also introduce some other new conditions which also characterize the above three spaces, that is, the conditions (M2), (M4) and (L2)-(L4) (see section 2 for the definitions). The condition (M2) is a variation of (M3) given in [3]. But it seems to be more natural than (M3) in connection with (Ml). The conditions (M4) and (L4) are closely related to Helgason's expansion in [8: p. 435]. Indeed, the above three mean values coincide with Helgason's, if the manifold is an Euclidean space or a globally symmetric space of rank one, and we are interested in to what extent the Laplacian determines the mean values. Our results include the assertion; one of the operators induced by the three mean values is expanded by means of a sequence of polynomials of Laplacian if and only if the manifold is harmonic (Theorem 2 (1)). As a by-product, we also obtain some sufficientconditions for a C6 manifold to be analytic (Theorem 1). In the course of our proof, we partially answer for smooth manifolds to Kowalski's conjecture given in [10] (Theorem 3).