ON THE SECOND CURVATURE OF A SUBPATH AND A UNION CURVE

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_{T．ADATI AND K． MATUMOTO}

ON T且E SECOND CURVATURE OF A SUBPATH

AND A UNION CURVE

BY

TYuzI ADATI AND K6zi MATUMOTO

In a Riemannian space we have THEoREM 1． The necessary and su伍cient condition that the s㏄ond curvature of

a脚desic皿ahypersu血ce in a Riemannian space be zero is that the geodesic

becomes a line of curvature on the hypersurface． As a generalization of this theorem we get THEoREM・2． The皿ecessary and su丘icient condition that the second curVature of asubpath with respect to a v㏄tor field in a Riemannian space be zero is that the v㏄tor field becomes a torse−forming vector along the curve． Next， let C be a union curve with respect tO a unit vector field La on a hyper− surface． Denoting the unit s㏄ond normal， the 丘rst and s㏄ond curvature in the enveloping space of C byξ（3）λ，κ（、），κ（，）， we get rc（1）＝Kn sinφsecθ，・②ξ・…一｛・…C・・ec・φ一（…φ）・｝芸λ＋｛（…㏄φ）・一・、・・…φ…㏄φ｝・・＋…ecφδ嵩λ， where Kn is a normal curvature of the hypersu血ce f（）r the dir㏄tion of C，θthe angle between the vector Lλand the normal of the hypersur垣Ce，φthe angle between

・he v㏄・・…and・h・…g・n・vec…芸λ・・C・』 ’r

From the above equations we have THEoREM 3． The s㏄ond curvature of a union curve with respect to a unit v㏄tor field La on a hypersurface in a Riemannian space is deter茄ined by the dir㏄tion of the curve and the vector LA．