CLOSED GEODESICS ON RIEMANNIAN MANIFOLDS
ï
As a preparation for the introduction of the structure of a differentiable
manifold on H\l, Ì) , modelled on a Hubert space, we first consider the
infinitesimal approximation of Ã'°°( Ë Ì) a t c G
C/GC(I,
M) given by the vector
space of piecewise differentiable vector fields along c. This vector Space can be
viewed as the 'tangent Space'
TcCf0C(L
M) of the 'manifold'
C/oc(7,
M) at the
point c. Our main interest lies in its completion with respect to an i/'-norm. To
get a precise formulation, we take TcC,0O{L Ì ) to be the space of sections in the
induced bündle c*r.
1.4 DEFINITION. Let c e C/oc(7, M).
(i) Define c*r to be the induced bündle over I:
c*TM - ^ TM
C*T Ô
/ - Ì
For each closed subinterval Ij C I with c \ Iy differentiable, the restriction of c*r
to 7y is the (differentiable) induced bündle. The fibre c*r~l(t) over t is also
denoted by TCJ.
(ii) By Cfx(c*TM) we denote the vector space of piecewise differentiable
sections of C*T. We also write TcCf0C(L M) instead and call this the tangent space
toC,00(7, M)atc .
(iii) Using the scalar product (, on the fibres TCJ of C*T, stemming from the
Riemannian metric on the corresponding TiV)M, we define, for £, ç in C'°°(c*TM):
(a) ll€IL = sup|€(')l·
/ e /
(b) ano=/#€(o^(o^
(c) €.ç é = ^ ï + í ß , í ç ï .
The norm derived from the scalar product ( , )
r
is denoted by || ||
r
, r = 0,1.
(iv) The completion of Cf0C(c*TM) with respect to the norms || H^ and || ||r is
denoted by C°(c*TM) and H\c*TM\ r = 0,1, respectively.
REMARK.
Á piecewise differentiable section of
C*T
is a continuous map ae : / -*
c*TM with c*r ï î = id and such that there exists a subdivision of / into closed
intervals J with c \ J differentiable and ae \ J differentiable. For î \ J we have the
covariant derivative V i | / with respect to the induced connection on
C*T\J.
Alternatively, î , or rather T*C Ï £, can be viewed as a piecewise differentiable
vector field along c.
Note that C°(c*TM) is a Banach Space, whereas
Hr(c*TM%
r = 0,1, are
Hubert Spaces.
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