On a new normalization for tractor covariant derivatives
On a new normalization for tractor covariant derivatives
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关于拖拉机协变导数的新归一化
DOI：
10.4171/jems/349
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发表时间：
2010
影响因子：
2.6
通讯作者：
J. Šilhán
中科院分区：
文献类型：
--
作者：
M. Hammerl;P. Somberg;V. Souček;J. Šilhán
流形M上G/P型正则正规抛物几何
产生不变微分的序列$D_i$
运算符，被称为弯曲版本的BGG分辨率。
这些序列是从正常的协变
在相应的牵引丛上的导数$\nabla^\omega$
其中$\omega$是标准的Cartan连接。第一
序列中的算子$D_0$是超定的，
已知$\nabla^\omega$产生这个的延拓
算子在齐次情况下M = G/P。我们的第一个主要结果
是这种延长的弯曲版本。这需要一个
V上拖拉机协变导数的新归一化。
此外，我们还得到了高阶算子$D_i$的一个类似结果.在
在这种情况下，需要修改外部协变导数，
$d^{\nabla^\omega}$的微分项。最后我们
用简单的射影例子说明这些结果，
共形几何和格拉斯曼几何。我们的方法是基于
标准BGG技术。
A regular normal parabolic geometry of type G/P on a manifold M
gives rise to sequences $D_i$ of invariant differential
operators, known as the curved version of the BGG resolution.
These sequences are constructed from the normal covariant
derivative $\nabla^\omega$ on the corresponding tractor bundle
V, where $\omega$ is the normal Cartan connection. The first
operator $D_0$ in the sequence is overdetermined and it is well
known that $\nabla^\omega$ yields the prolongation of this
operator in the homogeneous case M = G/P. Our first main result
is the curved version of such a prolongation. This requires a
new normalization of the tractor covariant derivative on V.
Moreover, we obtain an analogue for higher operators $D_i$. In
that case one needs to modify the exterior covariant derivative
$d^{\nabla^\omega}$ by differential terms. Finally we
illustrate these results with simple examples in projective,
conformal and Grassmannian geometry. Our approach is based on
standard BGG techniques.