334 K. IMPLICIT FUNCTION THEOREMLet (Mn,gM) and (Nm,gN) be Riemannian manifolds. Let (U, {xir= 1 ) be a
coordinate neighborhood of p E M and let (V, {ye} ;: 1 ) be a coordinate neighbor-
hood of q EN. Given k;:::: 1, we say that pointed maps F , G : (U,p) ---"* (V, q) have
the same k-jet (at p) if and only if
for all multi-indices a = (a 1 , ... , an) with 0 ::; lal ~ a1 + · · · +an ::; k and
1 ::; e ::; m, where pe ~ ye o F and ce ~ ye o G. Having the same k-jet at a
point p is an equivalence relation whose definition is independent of the choice of
coordinate systems in neighborhoods of p and q.Define l;,q (M, N) to be the set of k-jet equivalence classes [FJ~,q of maps F sat-
isfying F (p) = q. We define the vector bundle of k-jets of maps lk (M,N)---"*
M x N as the unionlk (M,N) ~ LJ l;,q (M,N).
pEM, qENkThe fiber l;,q (M, N) is isomorphic to EB ( S~M 0 TqN), where S i M denotes the
i=l
vector bundle of symmetric covariant i-tensors. A coordinate dependent vector
kspace isomorphism from l;,q (M,N) to EB (S~M 0 TqN) is given by
i=lNow we consider a coordinate-free point of view. Recall that dF is a section of
T* M 0 F*TN. Let(K.12) v = '\1^9 ,h : 000 (T M 0 FT N) ---" 000 (T M 0 T M 0 F T N)
denote the covariant derivative induced by '\J9 and F*'\lh. Define Vi= ('\19,hf ~
99,h o · · · o 99,h to be the i-th covariant derivative acting on sections of this bundle.
For example, '\lidF(p) E ®i+ir;M ®TF(p)N· Two maps Fang G have the same
k-jet at p if and only if they have the same value at p and the same higher covariantderivatives at p; i.e., F (p) = G (p) and '\lidF (p) = '\li dG (p) for O::; i::; k - 1.
Recall that the symmetrization of an element 0 E Q9 j r; M 0 TqN is given by
where the summation is taken over all permutations a. By standard commutator
formulas, the tensor 9i-^1 dF (p) is uniquely determined by the curvature tensor
and the symmetrizations of VJ-^1 dF (p) for all j ::; i. For this reason there is a