- CRITICAL POINT THEORY AND PROPERTIES OF DISTANCE SPHERES 479
EXERCISE I.38. Under the hypotheses in the proof of Lemma I.37 above,
we have
(I.37)(For a proof see for example Fact l.3(ii) of [107].)4.4. Distance spheres as Lipschitz hypersurfaces.
Since the distance function is only Lipschitz and the sphere S (p, s) is
not necessarily a smooth manifold, we recall the following.
DEFINITION I.39 (Lipschitz hypersurface). A Lipschitz hypersurface
in a smooth manifold is a subset which is locally the graph of a Lipschitz
function. That is, a subset N C Mn is a Lipschitz hypersurface if for every
q E N there exists a smooth local coordinate chart (U, x = {xi}) with q E U
and x(U) = V x (a, b), where V c ffi.n-l is an open set and -oo <a< b < oo
and where x (N n U) is the graph of a Lipschitz function on V.
If N is a connected Lipschitz hypersurface of a complete Riemannian
manifold (Mn, g), then any two points x, y E N may be connected by a
Lipschitz path in N,^10 which in turn has finite length. Hence the induced
distance dN ~ d.cdN' as defined in (G.6), is finite.
We now show the following.
LEMMA 1.40. If N is a connected Lipschitz hypersurface of a complete
Riemannian manifold (Mn, g) such that N is a closed subset of M, then
the length space N is complete.
PROOF. Given x, y EN, let 'Yi : [O, 1] -+ N, i EN, be a length minimiz-
ing sequence of constant speed paths joining x to y, i.e., ,C ('Yi) -+ dN (x, y)
and for any s1, s2 E [O, 1] we have
,C ( 'Yil[s 1 ,s 2 ]) = ,C ('Yi) ls1 - s2I ·Hence
dM ('Yi (s1), 'Yi (s2)) < ,C ( ·) < C
is1 - s2I - 'Yi -for s1 =f. s2 and where C < oo is independent of i, s1, s2. By the Arzela-
Ascoli theorem, there exists a subsequence such that 'Yi converges to a path'Y=: [0,1]-+ N joining x toy. By definition, dN(x,y) :S £("!=)· On the
other hand, by the lower semi-continuity of £, we have,C ('Yoo) :S ,lim ,C ('Yi)= dN (x, y),
i--700so that ,C ('Yoo)= dN (x, y).^0(^10) Locally, such a path may be obtained by intersecting N with a smooth transversal
surface.