## Access

You are not currently logged in.

No Cover Image

# Lie Equations, Vol. I: General Theory. (AM-73)

ANTONIO KUMPERA
DONALD SPENCER
Pages: 309
Stable URL: http://www.jstor.org/stable/j.ctt1b9rzwm

1. Front Matter (pp. i-iv)
2. FOREWORD (pp. v-viii)

In his papers [13(a), (b)], Spencer developed a general mechanism for the local deformation of structures on manifolds defined by transitive continuous pseudogroups. A new version of this theory, based on the differential calculus in the analytic spaces of Grothendieck, has been given by Malgrange [9(c)] in his proof of the integrability (existence of local coordinates) of almost-structures defined by elliptic transitive continuous pseudogroups (or elliptic Lie equations), under a certain integrability condition.

The authors here redevelop the theory by two different approaches. The starting point of one approach is based on the idea of B. Malgrange in which the...

3. GLOSSARY OF SYMBOLS (pp. ix-xii)
5. INTRODUCTION (pp. 3-48)

A s the title indicates, the content of these notes is a lengthy construction of techniques devised to study specific differential geometric problems some of which we expect to treat in Part II. In this introduction we state our main objectives and illustrate by examples some of their geometric implications. A fairly detailed summary of the present monograph is given in [43]. The references will provide further motivation.

The main goal of these notes is the construction of the non-linear complexes (23.9)k+1, (25.3)k+1and (30.1)k+1as well as their linearizations (6-2)k+1, (7.1)k+1and the first two lines of (29.6)k(more...

6. CHAPTER I JET SHEAVES AND DIFFERENTIAL EQUATIONS (pp. 49-87)

Throughout these notes X w ill be a Hausdorff C-manifold of dimension n anddifferentiablewill mean of class C. Since the results are local, one can replace X by an open set in Rn. Let E→X be a vector bundle (always locally trivial, Cand of finite rank) and E the sheaf of germs of sections of E. We recall that local sections of E identify with local sections of E. For the trivial line bundle E = X x R we obtain the structure sheaf E = 0x(or simply 0 ) of the the manifold X....

7. CHAPTER II LINEAR LIE EQUATIONS (pp. 88-103)

The L ie bracket of germs of vertical vector fields (elements of Tv) is a left θ-bilinear operation, hence defines a left θ-lin e a r map

$[,]:{ \wedge ^2}\vartheta {T_V} \to {T_v}$

(exterior product with respect to the left structure of Tv). Since the Lie derivative L(ξ) , ξε Tx², satisfies$L\left( \xi \right)\left( {{g^{k + 1}}} \right) \subset {g^k}$, it follows that the above bracket factors to a left θ-linear map

$\left[ , \right]:{ \wedge ^2}\vartheta {I_k}T \to {I_{k - 1}}T$

(recall that${I_k}T = {T_v}/{g^{k + 1}}{T_v} \simeq {I_k}{ \otimes _\vartheta }T$). This latter bracket is transformed by II₂ (cf. Section 4) into a sheaf map which is the extension, to

germs, of the bundle map

$\left[ , \right]:{ \wedge ^2}{J_k}T \to {J_{k - 1}}T$

defined by$\left[ {{j_k}\xi \left( x \right),{j_k}\eta \left( x \right)} \right] = {j_{k - 1}}\left( {\left[ {\xi ,\eta } \right]} \right)\left( x \right)$where ξ,η are vector fields on X. We...

8. CHAPTER III DERIVATIONS AND BRACKETS (pp. 104-135)

In this section we shall recall some basic facts about derivations of scalar differential forms and make explicit the notations. For a detailed account we refer the reader to [3] and [4]. L et M, N be modules over the same ring and$u:{ \wedge ^r}M \to M,V:{ \wedge ^s}M \to N$skew-symmetric forms of degree r and s respectively. Define$i\left( u \right)v = u \wedge v:{ \wedge ^t}M \to N, t = r + s - 1$by

$u\bar \wedge v\left( {{x_1},...,{x_t}} \right) = \frac{1}{{r!\left( {s - 1!} \right)}}\sum \in \left( \sigma \right)v\left( {u\left( {{x_{\sigma \left( 1 \right)}},...,{x_{\sigma \left( r \right)}}} \right),{x_{\sigma \left( {r + 1} \right)}},...,{x_{\sigma \left( t \right)}}} \right)$

where divisibility assumptions are made on the ring (which in fact is irrelevant, cf. [3], [4]). If u and v are decomposable, namely if$u = a \otimes \xi \in \left( {{ \wedge ^r}M} \right) * \otimes M,V = \beta \otimes \eta \in \left( {{ \wedge ^s}M} \right) * \otimes N$, then$u\bar \wedge v = \left( {a \wedge i\left( \xi \right)\beta } \right) \otimes \eta = \left( {u\bar \wedge \beta } \right) \otimes \eta$where i(ξ) is the usual interior product.

Let X be a differentiable manifold, f(X) the Cfunctions...

9. CHAPTER IV NON-LINEAR COMPLEXES (pp. 136-211)

L et X be a manifold. Denote by CX the sheaf of germs of local C-maps of X and by But X the sheaf of germs of local C-diffeomorphisms of X. L et JKX be the manifold of k-jets of local C-maps f: U -» X, U open in X. Thesourcemap

${\alpha _k}:{J_k}X \to X,{j_k}f\left( x \right) \to x$,

and thetargetmap

${\beta _k}:{J_k}X \to X,{j_k}f\left( x \right) \to f\left( x \right)$,

are submersions onto X. JkX has a natural structure of small differentiable category, the composition being defined by

${j_k}g\left( y \right).{j_k}f\left( x \right) = {j_k}\left( {g \circ f} \right)\left( x \right)$

with y = f(x). The units are the elements jkId(x) which can be identified with the points of X. Let IIkX be...

10. CHAPTER V DERIVATIONS OF JET FORMS (pp. 212-256)

The purpose of this chapter is to extend the Frölicher-Nijenhuis theory (Section 13) to the sheaf

$\wedge \left( {{{\bar I}_k}T} \right) * { \otimes _\vartheta }{I_k}$

and transcribe into this context some of the previous results, in particular, the linear and non-linear complexes.

Recall that Ikis a left and right θ-algebra and that ĪkT is a module over the algebra Ikhence ĪkT also carries a left and right θ-module structure. Let (ĪkT)* be the dual of ĪkT with respect to the left θ-module structure. (ĪkT)* the sheaf of germs of (left) θ-linear 1-forms ĪkT→θ. Since ĪkT is locally free of finite rank, the stalk of (ĪkT)* at...

11. APPENDIX LIE GROUPOIDS (pp. 257-277)
12. REFERENCES (pp. 278-285)
13. INDEX (pp. 286-295)