152 
This is the Gaussian curvature theorem, generalized for Vn in V, *). 
When we write 
2 5 
63 Vi, heikele he he, (38) 
we can give to (37) on account of (17) the following form: 
Bu 
EEKSK — 2 2 (he hin. -* he) (39) 
_ The second term on the right in (37) and (39) vanishes e.g., when 
the V,, is geodesic in P. Hence the theorem holds: 
The Rremann-Curistorven affinor of a Vy in Vo, geodesic in a 
point, is in this point the V,,-component of the RimmMann-CurisTOFFEL 
affinor of the Vn. 
The second term on the right vanishes also in a V,, with only 
axial points, when the degree of nullity of the tensor *h, belonging 
to the favoured normal direction, is one. 
If the V, is an S, (comp. p. 146): 
4 
K = 2 K, *g = *g’) (40) 
‘and the J,, is ARE ANDES the term on the left in (37) and (39) 
passes into 
we 2 KBA | (41) 
Hence 2(H ~ ’H,)(H, ~ ’H,) H,.’H, in this case depends only on 
the linear element of V,,, the m-direction of the Vin in the considered 
point and on K,. If the V, is an S, and if the Vs, ‘contains only 
umbilical points, we derive from (39) that the V,,is an S,. This holds 
consequently also in the particular case that the V,, is geodesic. 
§ 5. Absolute, relative and forced curvature of a Vm in Vn. 
1 
Transvecting (39) totally with — Fie a’b’b’a’, we get: 
des, yer dt 2 
US OP eg shit Pia 4S ~'h,) (he —'he). (44 
mm DE REEF jd ST de) emit). (42) 
1 e 
We call XK, = — ———-a'b'b'a'! K the absolute curvature of 
m (m—1) 
the V, it is the curvature of V,, considered as a manifold for its 
1) Lipscuitz, 70, 1 p. 292, Vm in Vn; Voss, 80, 1 p. 139, Vm in Vy; Rica, 
02, 1, p. 859, Vm in V„; Kürne, 03, 1, p. 309, Vm in Va; comp. also e.g. BrancHt 
99,1, p. 602, Vn: in Vy; SERVANT, 02, 3, p. 94, Vs in Ry. 
4) 21, 1 p. 76. 
