Mathematics — “On Curvature and Invariants of Deformation of 
a Vaan V,.” By Prof. J. A. Schouten and D. J. Srruix. 
(Communicated by Prof. Jan pe Vaixs). 
(Communicated at the meeting of October 29, 1921). 
$ 1. Antroduction. 
In the theory of curvature of a V,,, imbedded in a V,, a great 
number of. theorems can be deduced with the aid of the quantity 
3 
of order three H without use of the Rimmann-CurisToFFEL quantity 
of order four of the Vs and the V,, K and K. We have developed 
these theorems in detail in another paper’) and we will indicate 
them here only as far as is absolutely necessary. Other theorems, 
e.g. those concerning the invariants of deformation, depend on the 
4 4 
mutual relations of the quantities K and K'. This paper will treat 
the most important theorems of this kind for the most general case, 
which is not yet investigated sufficiently. 
§ 2. Vin in Vy, absolute, relative and normal curvature of a 
congruence. 
Suppose a congruence i*) in a JV,, in V,. The fundamental tensor 
of the V, be *g—aa=bb—..., the fundamental tensor of the 
Vn being *g’ = a'a’ — b’b’ — ze. “lm, the,.V,,,.eneess ae gunuual 
orthogonal congruences i,,...,i,, in such a way that V,, is built-up 
by curves i,,...,i,. We suppose that the suffixes 2, 7,4,/ may get 
the values 1,,...,n; @,6,¢,d dhe values.din. vern Me and,é./ 9, 0 
the values m+1,...,n. 
When 
a’ = > ae ie (1) 
e 
hence 
1) Vp means in this paper a p-dimensional manifold, whose linear element is 
represented by the square root of a general quadratic differential form, Sp means 
such a manifold with constant RiEMANN-curvature, R, such a manifold with 
euclidean linear element. 
4) 22.1. 
’) The notations used in this paper are developed in detail in 21. 1, shorter 
also in 21.2. 
