论文标题
曲率适应的半曼尼亚人群
Curvature-adapted submanifolds of semi-Riemannian groups
论文作者
论文摘要
我们研究了配备双重视度的谎言组$ g $中的任意consimensions的半帝国次级submanifolds。特别是，我们表明，如果$ M \子集G $的普通束在Lie支架下关闭，则任何普通的Jacobi操作员$ K $ of $ m $ of $ m $等于相关的不变形状运算符$α$的平方。这允许从左翻译中理解$ g $的曲率适应性。例如，如果$ m $是Riemannian Hypersurface的情况，我们的主要结果指出，正常的Jacobi操作员与普通形状操作员的通勤正是当其每个特征空间的剩余不变性扩展与其他所有特征的一阶相切相切时，则与其他所有特征相切。作为平等$ k =α^{2} $的进一步结果，我们获得了一个众所周知的事实的新的独立于案例的证据：每个配备了双重不变的半 - 雷曼式指标的三维谎言组具有恒定的曲率。
We study semi-Riemannian submanifolds of arbitrary codimension in a Lie group $G$ equipped with a bi-invariant metric. In particular, we show that, if the normal bundle of $M \subset G$ is closed under the Lie bracket, then any normal Jacobi operator $K$ of $M$ equals the square of the associated invariant shape operator $α$. This permits to understand curvature adaptedness to $G$ geometrically, in terms of left translations. For example, in the case where $M$ is a Riemannian hypersurface, our main result states that the normal Jacobi operator commutes with the ordinary shape operator precisely when the left-invariant extension of each of its eigenspaces has first-order tangency with $M$ along all the others. As a further consequence of the equality $K = α^{2}$, we obtain a new case-independent proof of a well-known fact: every three-dimensional Lie group equipped with a bi-invariant semi-Riemannian metric has constant curvature.
