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.