Flat totally geodesic submanifolds of quasisymmetric Siegel domains
Flat totally geodesic submanifolds of quasisymmetric Siegel domains
复制标题
拟对称 Siegel 域的平坦全测地线子流形
DOI：
10.1007/bf00182400
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发表时间：
1988
影响因子：
0.5
通讯作者：
J. Dorfmeister
中科院分区：
文献类型：
--
作者：
J. D'Atri;J. Dorfmeister
具有Bergman度量的有界齐性域（等价于齐性Siegel域）构成了一大类Kihler-Einstein流形.其中，有界对称域产生了许多有趣的微分几何结果。更大类的拟对称域应该表现出非常有趣的微分几何特征，特别是应该表明在一般情况下人们可以期望什么.本文引入了“典型平面”（全测地子流形）的概念，并刻画了它的切方向.它们（本质上）是正则真实的对称子流形的平坦全测地子流形的方向。对于拟对称域，我们表明，这些方向是方向的协变导数零化的曲率张量，Vw R = 0。后一个结果的证明使用并推广了[9]中关于拟对称Siegel域分类的证明。
Bounded homogeneous domains (equivalently, homogeneous Siegel domains) equipped with the Bergman metric form a large class of K~ ihler-Einstein manifolds. Among them the bounded symmetric domains have generated many interesting differential geometric results. The larger class of quasisymmetric domains should exhibit comparably interesting differential geometric features and in particular should indicate what one can expect in the general case.In this note we introduce the notion of a'canonical fiat'(totally geodesic submanifold) and characterize its tangent directions. They turn out to be (essentially) the directions of a flat totally geodesic submanifold of a canonical real symmetric submanifold. For quasisymmetric domains we show that these directions are the directions for which the covariant derivative annihilates the curvature tensor, Vw R= 0. The proof of this latter result uses and generalizes (some of) the proofs given in [9] for the classification of quasisymmetric Siegel domains.