Cartan引理的推广

本文将现代微分几何中著名的Cartan引理,推广到二次外形式空间(?)~2(V~*),得出两个定理.

Generalization of Cartan's Lemma

In this paper, we generalize the Cartan's lemma to ∧2(v*), and obtain. Theorem 1, Let w1, w2, …, wr, wr+1, …, wn be a basis and θ1, θ2, …,θr∈∧2(v*), (1≤r≤n), then ∑r wi/θi= 0 if and only if there areauvw∈F, such that where a jkj + aijk- aikj = 0(1≤i,j,k≤r) and Theorem 2, Let w1, w2, …, wr∈v*(1≤r≤n) be r-linear independent pfaff forms and g∈∧2(v*) then that pfaff equation system wi= 0(i = l, 2, …, r) implys g= 0 if and only if there are θ1, θ2 , …, θx∈v* such tha