By P. Ciarlet

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**Additional resources for An Intro. to Differential Geometry With Applns to Elasticity**

**Example text**

It has to be taken into account to prevent a confusion when the commuting property is considered for the elements of abelian group PGalab. 1 contains a lot of intermediate lemmas and all this paragraph will be devoted to it. I start with the following geometric observation: for anyone-dimensional subfield J{' of J{ the group PGal cJ{' is a free central extension of the abelian group PGalab J{' and therefore the images of commuting elements x, y have to be proportional in PGalab J{'. That means x, yare in fact proportional Zp-functions being restricted on K' as a subspace of K.

It can't happen because we can find V2 intersecting all the spaces Va; in general stratum. Thus for any four points on pI the function x takes the same values at three of them. Therefore x on pI is either constant or delta-function. If it is constant for any line pI in P(L*) then it is constant also on P(L*). If it is a delta-function on pI of the point (x) then it will be the delta-function of the same one point on P(L*). Indeed for any other projective line pI, we can find two-dimensional subspace intersecting all Va, s in pll.

The latter one is generated by the elements of type jI/m, where f in m-th power locally everywhere. We denote the subgroup Z(5)/PicX by G z . This notation assumes that Gz does not depend on X, but it is clear from the above duality argument. Now I want to restore Gz in terms of the PGaZ c , but first I would like to define also GZ(p) = Gz ® Z(p), where Z(p) C Q is a sub ring of p-integer rational numbers. The group GZ(p) is Z(p) sublattice of PGazab. REMARK: By assumption on K, (PicX is free and finitely generated abelian group), I identify PGazab with a quotient of Zp(5) by PicX ® Zp-finite dimensional Zp-subspace with a lattice.

### An Intro. to Differential Geometry With Applns to Elasticity by P. Ciarlet

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