150 
is the second fundamental tensor of the V,-1*). It is obvious that 
in general 
Vin = 7h + Li tins (23) 
where u,=—i,1!Vi, is the curvature vector of i,. If we choose 
i, geodesic, the formula is simpler: 
oh = ds (24) 
3 
Aa, SAV LE: (25) 
The forced curvature vector 
u'—=— ii? hi, (26) 
has for m=n—1 always the direction of i, and gets an extreme value 
in the principal directions of *h. Hence these principal directions are 
also the directions of principal curvature’). When i,,a=1,...,n—1 
are unit vectors in these directions and Zi, the principal radii of 
curvature, with positive sign when the curvature vector has the sense 
of i,, we have 
1 
arent 4 
h=— = lee (27) 
and from this we get the theorem: 
The degree of nullity*) of the second fundamental tensor is equal 
to the number of vanishing principal radi of curvature. 
If all principal radii of curvature in a point of the V,_; are 
equal, and only in this case, *h is a multiple of the fundamental 
tensor of the Vn: 
1 
h=— = (28) 
3 7 
and H thus assumes the form (20). Hence the point is an umbilical point. 
§ 4. Relations between the RinMANN-CHRISTOFFEL affinors 
of the Vn and the Vy. 
For the V,, the RrEMANN-Curistorrer, affinor has the form ‘*): 
4 
K= 2 AV Nja ae {Vis => Wibe) | @— dD) (29) 
1) We can find indeed: h,,= !/s in 1 Vg), Compare e.g. Biancm 99. 1, p. 601 
and ScHOUTEN—STRUIK 19, 1, p. 207; 19. 2, p. 601. 
5) First defined by KRONECKER 69, 1, Vn—1 in Rp. 
5) The degree of nullity of a tensor of second order in Vp is the degree of 
nullity of the matrix of the p? components. 
4) Comp. 21.1, p. 73. 
