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2. Basic Calculus Lemma and the existence of d:

In the following lemma J would denote an ordered k−set {j1, j2,...jk}, 1 ≤ j1 < j2 < ··· < jk ≤ n. We shall consider pairs (s,J) such that 1 ≤ s ≤ n and s =6 j1, j2,...,jk. Let N be the total number ′ ′ of such pairs. For a given pair (s,J) and jp ∈ J denote by (jp,J ) the complementary pair (jp,J ) obtained by removing jp from J and inserting s in the right place.

Lemma: Let φ1,φ2,...,φk be k smooth functions of z1, z2,...,zn. Then

∂ ∂(φ1,φ2,...,φk) dzs ∧ dzj1 ∧···∧ dzjk =0. X ∂zs ∂(zj1 , zj2 ,...,zj ) s,J k

Proof: Assume jq−1

2 ∂ φ1 ∂φ1 ∂φ1 ∂z ∂z ∂z ... ∂z s jp j1 jk (−1)p−1 ...... 2 ∂ φ1 ∂φ1 ∂φ1 ∂z ∂z ∂z ... ∂z s jp j1 jk

Together with the differentials, we get the monomial:

2 ∂ φ1 ∂φ1 ∂φ1 ∂z ∂z ∂z ... ∂z s jp j1 jk p+q−2 (−1) ...... dzj1 ∧···∧ dzjq−1 ∧ dzs ∧ dzjq ∧···∧ dzjk (1) 2 ∂ φ1 ∂φ1 ∂φ1 ∂z ∂z ∂z ... ∂z s jp j1 jk

′ Now we consider the term arising out of the complementary pair (jp,J ) and look at the relevant monomial namely the one in which the second derivatives

∂2φ i , i =1, 2,...,k ∂zjp ∂zs

2 appear in the determinant. These second derivatives appear in the qth column (jq−1

dzjp ∧ dzj1 ∧···∧ dzjq−1 ∧ dzs ∧ dzjq ∧···∧ dzp−1 ∧ dzp+1 ∧···∧ dzjk , (s < jp).

Owing to the presence of dzs, the dzjp has to now move through p transpositions to get this in standard form. Thus we get the term (1) but with (−1)p+q−1 instead. Thus the terms arising from complementary pairs cancel out. The proof is complete.

Definition (The exterior derivative d): The standard notation for the set of all smooth k-forms on M is Ωk(M). We introduce the R-linear map

d : Ωk(M) −→ Ωk+1(M).

Let ω be a differential k form and on the chart U let ω be given by

U ω = a dxi1 ∧···∧ dxi X i1i2...ik k i We define on each chart U dω = da ∧ dxi1 ∧···∧ dxi . (2) X i1i2...ik k i

The notation stands for the sum over all standard k−tuples (i1, i2,...,i ) with i1 < i2 < ··· < i . X k k i The basic properties of this operator can almost be read off from this definition except for one hurdle. We do have the job of showing that d is well defined namely, of verifying consistency on overlapping charts but we shall demonstrate that this is nothing but the basic calculus lemma!

Theorem: The operator d given by (2) is a well-defined element of Ωk+1(M).

Proof: Let U and V be two overlapping charts. Need to check that

U V da ∧ dxi1 ∧···∧ dxi = da ∧ dyj1 ∧···∧ dyj , on U ∩ V. X i1i2...ik k X j1j2...jk k i j

Since ∂(x 1 , x 2 ,...,x ) aV aU i i ik j1j2...jk = i1i2...ik (3) X ∂(y 1 ,y 2 ,...,y ) i j j jk our job is to check that

U U ∂(xi1 , xi2 ,...,xik ) da ∧ dxi1 ∧···∧ dxi = d a ∧ dyj1 ∧···∧ dyj i1i2...ik k  i1i2...ik  k X X X ∂(y 1 ,y 2 ,...,y ) i i j j j jk

Well, the right hand side breaks up into two sums:

U ∂(xi1 , xi2 ,...,xik ) d a ∧ dyj1 ∧···∧ dyj = I + II.  i1i2...ik  k X X ∂(y 1 ,y 2 ,...,y ) i j j j jk

3 The first sum I displayed below is tensorial namely,

U ∂(xi1 , xi2 ,...,xik ) U da ∧ dyj1 ∧···∧ dyj = da ∧ dxi1 ∧···∧ dxi i1i2...ik   k i1i2...ik k X X ∂(y 1 ,y 2 ,...,y ) X i j j j jk i which is the desired result and so we must show that the second (non-tensorial) term II is identically zero namely,

U ∂(xi1 , xi2 ,...,xik ) II = a d ∧ dyj1 ∧···∧ dyj =0. i1i2...ik   k X X ∂(y 1 ,y 2 ,...,y ) i j j j jk Since the coefficients, apart (3), are arbitrary smooth functions, we must show that each of the pieces

∂(xi1 , xi2 ,...,xik ) d ∧ dy 1 ∧···∧ dy   j jk X ∂(y 1 ,y 2 ,...,y ) j j j jk individually vanishes. That is to say for each fixed i1 < i2 < ··· < ik we have

∂ ∂(xi1 , xi2 ,...,xik ) dy ∧ dy 1 ∧···∧ dy =0.   s j jk X ∂y ∂(y 1 ,y 2 ,...,y ) j s j j jk

But this exactly the calculus lemma. The proof is complete.

References

[1] S. S. Chern, W. H. Chen and K. S. Lam, Lectures in differential geometry, World Scientific 2000.

[2] Hicks, Notes on differential geometry, Van Nostrand, New York, 1965.

[3] J. R. Munkres, Analysis on manifolds, Addison-Wesley Publishing Co., 1991.

[4] H. Samelson, Differential forms, the early days; or the stories of Deahnah’s theorem and Volterra’s theorem, American Mathematical Monthly, 108 (2001) 522-530.

[5] M. Spivak, Calculus on manifolds, Addison-Wesley publishing co., Reading, Massachusetts, 1965.

[6] M. Spivak, Comprehensive introduction to differential geometry, volume 1, Publish or Perish, Inc., Houston, Texas, 1999.

[7] M. Spivak, Comprehensive introduction to differential geometry, volume 2, Publish or Perish, Inc., Houston, Texas, 1999.

[8] T. J. Willmore, Riemannian Geometry, Clarendon Press, Oxford (2002).

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