- HOLDER SPACES AND SOBOLEV SPACES ON MANIFOLDS 335
natural bundle isomorphism defined by
kJ: Jk (M,N)---+ EB (piSiM @p2TN)'
i=l
k
J([FJ;,q) =EB Sym (vi-^1 dF) (p),
i=l
where PI : M x N---+ M and P2 : M x N---+ N denote projections.
Each vector bundle Pi Si M 0 p?_T N over M x N has a natural inner product
9i on each fiber induced by 9M and 9N and also has a natural connection (i)V
induced by the Riemannian connections \J9M and \J9./\f. We obtain a fiber metricand compatible connection on the vector bundle Jk (M,N) via pull-back by J.
The connection defines parallel translation and horizontal spaces for Jk (M, N).
So, by a general construction, we obtain a unique Riemannian metric 9Jk on thetotal space Jk (M, N) with the following properties. The restriction of 9Jk to the
fibers is the fiber metric on Jk (M,N), the horizontal spaces are g 1 k-Orthogonal to
the fibers, and the 9Jk restricted to the horizontal spaces project isometrically to
9M + 9N on M x N.Given k;::: 1, let d 1 k be the Riemannian distance on Jk (M,N) associated to
the metric g 1 k. Given Ck maps F , G : M ---+ N, we may now define the Ck (M, N)
distance between them as
(K.13) dck(M,N) (F, G) ~ sup d 1 k ([FJ;,F(p)' [GJ;,G(p)).
pEMFor k = 0, we may define
(K.14) dco(M,N) (F, G) ~ sup d 9 ./\f (F (x), G (x)).
xEMThe metric space (Ck (M,N) ,dck(M,N)) is complete fork;::: 0.
Following Hamilton's presentation in §8 of [143] we may also consider jets of
mappings as homomorphisms of jets of paths. Given a smooth manifold P, letI = ( -1, 1) and define the vector bundle
Jkp = LJ J~,q (I, P) ---+ P.
qEPThe space Jkp is the quotient space of equivalence classes of paths a : I ---+ P h aving
the same k-jet at 0. Note that Jkp is naturally bundle isomorphic to EBk TP. Via
this isomorphism, given a Riemannian metric on P and its associated Levi-Civita
connection, we may define a Riemannian metric on the total space Jkp_ Hence we
may define the Riemannian distance d 1 kp between two k-jets.
Given k ;::: 1 and a Ck map F : M ---+ N, we define the induced map
Jk F : Jk M ---+ Jk Nby
Jk F([a]~,a(o)) = [F o a]~,F(a(O)).
We may also define the Ck·"'-distance between nearby hypersurfaces. Let
(Mn,g) be a Riemannian manifold with sectional curvature bounded above by
r;,^2 > 0. Let xn-^1 c M be a closed smooth hypersurface. Let v be a choice of
smooth unit normal vector field to X and assume that -A Ix :::; Ilx :::; A Ix, where