Stokes定理和光纤积分

student123 06/20/2018. 1 answers, 35 views
integration derivatives differential-geometry fiber-bundles stokes-theorem

我想了解以下内容:

让$ \ pi:X \到Y $一个光纤束和$ \ omega $一个封闭的平滑差分形式。 将$ I:Y \定义为\ mathbb C $,$ y \ mapsto I(y)= \ int _ {\ pi ^ { - 1}(y)} \ omega $。 那么斯托克定理暗示$ I $是常数。

这种情况出现在我想要理解的文本中。 有更多的假设(例如$ \ pi $本身),但我认为他们只需要达到这种情况。

到目前为止,我注意到如果$ \ pi ^ { - 1}(y)= \ partial M $,那么由Stoke的thm $$ \ int _ {\ pi ^ { - 1}(y)} \ omega = \ int_M d \ omega = 0。$$所以也许我需要将这个积分与$ dI $或$ \ frac d {dy} I(y)$联系起来,但我不知道该怎么做。

1 Answers


Ted Shifrin 06/20/2018.

因为$ \ pi $是正确的(是的,这非常重要),光纤$ \ pi ^ { - 1}(y)$是$ X $的紧凑$ k $ -dimensional子流形。 当然,您还需要在$ X $和$ Y $连接时添加$ \ omega $是一个已关闭的$ k $ -form。

假设$ y,y'\在Y $中,并通过平滑路径$ \ gamma $加入它们。 那么$ \ pi ^ { - 1}(\ gamma)$是一个紧凑的子流形$ W \子集X $,其中$ \ partial W = \ pi ^ { - 1}(y') - \ pi ^ { - 1}( Y)$。 你现在可以完成吗?


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