GENERALIZATION OF A THEOREM BY Po HARTMAN AND A. WINTNER By

Theorems of standard differential geometry are usually stated under strong sufficient conditions. In order to reduce conditions in the fundamental theorem of 3-dimensional space curves a modified treatment was given hy P. HARTl\IAN and A. WINTNER [1]. Their result will he generalized in this paper for higher dimensions. 1. Basic facts and definitions The foundation of differential geometry of curves in n-dimensional spaces was given hy W. BLASCHKE [2]. The definitions of higher curvatures,vill he reviewed here as they were introduced hy H. GLUCK [3] and [4] and earlier hut in a less concise form hy EGERVARY [5]. A hit of modification,vill he given here too hy introducing signed higher curvatures_ Let v-n he the oriented n-dimensional Euclidean vector space formed hy vectors of the n-dimensional Euclidean space En and AP ( vn) for p = 1, _. _, n the ('\dimensional Euclidean vector space formed hy the p-vectors over vn p,vith the inner product induced hy that of vn. Let x = x ( i) he a vector-valued function which represents a curve C c En in a well-known sense, i.e. x = x ( i) = @(i), where i runs over an interval Iof real numhers, P ( T) is the point on C which helongs to i and 0 E En is a fixed point considered as the origin of the vectors. The p-dimensional osculating suhspace Lp to the curve at a point P(T) is defined as spanned hy the linearly independent derivative vectors x(i) (T), i = 1,. _ _ , p, and the p-th higher curvature 'Y.,p, as the measure of the rate of turning of the appropriate p-dimensional osculating suhspaces,vith respect to the arc length where p = 1, __ _ , n- 1. This latter definition can he 'Nritten therefore in the follo,ving form 'Y.,p = 1 II~n ~,p=l, __ _,n-l Ilx'(i) 11 I d