調和葉狀結構--一個有趣的公式(觀點)

LiuH41發表於2024-07-11

\[E(\mathcal{F})=\frac{1}{2}||\pi||^2= \frac{1}{2} \int _M g_Q(\pi\wedge * \pi) \]

Theorem: Let \(\mathcal{F}\) be a foliation on a manifold \(M\) and \(g_M\) a Riemannian metric. Then all the leaves of the foliation are minimal submanifolds of \(M\) if and only if the canonical \(Q\) -valued 1-form $\pi : TM \to Q $ is harmonic.

Kamber, F.W., Tondeur, P. (1982). Harmonic foliations

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