Imbedding and Curvature
In a V n the length ds of d ξ x , called the linear element of the V n ,is given by
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§ t. The X 1 in V,..
227
§ 8. Simple and semi-simple groups. A group is called simple if it does not contain any invariant subgroup. It is called semi-simple if it does not contain any integrable invariant subgroup. Hence a simple group is also semi-simple. As the group of gba is an integrable invariant subgroup (cf. IV § 7) it follows that for a semi-simple group gba has rank r. Äs a corollary we get that the group space of a semi-simple group has a riemannian geometry. But this condition is not only necessary but also sufficient for the group to be semi-simple. We prove that the rank of gba is always
d"ll> { }
d,"+lrp
-d { s} = 0; ... ;-d,. s = 0 ; -u+J d{ s 0 s 0 s
(1.13)
s} =!= 0
0
there is said to be a contact of order u at the point of intersection ~" = 0
r (s). 0
If (1.12) is satisfied identically in s the curve lies on the hypersurface. If u = 1, the direction of the curve at ~" lies in the (n -1)-direction 0
of the hypersurface at that point. But if u = 2 the osculating R 2 does not necessarily lie in this (n -1)-direction and a similar remark can be made for all higher values of u.
§ 2. The X 1 in W.. and Lw 1) Curves in a space with a generallinear connexion can only be dealt with in a satisfactory manner if it is possible to define in some invariant way a parameter and a tangent vector. If no connexion is given in a space it is not possible to define an invariant parameter and this is the reason why for instance an xl in xn has no osculating 2-direction. For an X1 in Wn [cf. III § 3J2) the factor in g;,." can be fixed at a point ~" of the X 1 . Then this fixed g;,." can be displaced parallel 0
along the curve and at each point a tangent vector f" can be constructed having unit length with respect to this tensor field. The field f" is then fixed to within a constant scalar factor by the connexion in Wn and the curve. Taking f" = d~"jds and s = 0 for f' = ~" we get a parameter s 0
on the curve which is also fixed to within a constant scalar factor. This parameter s can now be used to establish FRENET formulae in exactly the same way as was described in the preceding section. From these formulae we see then that all vectors f" and all curvatures k are fixed to within s s the same constant scalar factor and that consequently the proportians 1)
Cf.
2)
HLAVATY
1935, 1928, 3.
HLAVATY
2,
I.
V. Imbedding and Curvature.
232
of the k and also the vectors k i;. and k- 1 i" are invariants of the curve s
1 s
s
1
and the connexion. 1 ) The case of an X 1 in Ln is much more complicated. 2) Let such a curve be given by its parametric equation
!;" = f"(t).
(2.1)
Then we may form the vectors
v" def d!;" .
(2.2)
1
Ö d!;".
>< R" vp).'"
+ 2l '"' hc]b [d
·
This is the generalized equation of GAuss for a rigged A,._ 1 in A,.. From (3.11) and (3.12) we get by differentiation and alternation (3.16}
2'll[ahclb=- 2B[d~JtV,.B~~V11 ta = 2B[d~Ji (V,. t;. na) V~' ta- 2B[d~Ji V. V~' t;. = - 2Bfd hc]b na VI' ta + B;t~g R;;.;." f,.;
1
(3.17)
Now in (3.15) all terms are invariant if the nor
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