Rigidity for isometric imbeddings
Eiji KANEDA, Noboru Tanaka · Kyoto journal of mathematics · 1978
io n o f a m a n ifo ld M in to th e m-dimensional E uclidean space R"'.Assume th at f is n o n -d e g e n e r a te (s e e § 1 ) .In h is p a p e r [1 5 ] on e of th e a u th o rs h a s sh o w n th a t th e re is a sso c ia te d to f a linear differential operator L in a natural m anner, and th a t it is equivalent, in a sense, to th e differential operator d O f o f infinitesimal isometric deformations o f f ([15], T heorem 1 .2; T h e o r e m 1. 1 in the present paper). E speciallyth e infinitesim al isom etric deform ations u o f f a re in a o n e-to -o n e co rresp o n d en ce w ith th e solu tion s yo of the equation Lço -= 0 .It sh o uld b e h ere p o in ted out th a t t h e sym bol of the operator dO, necessarily degenerates, w hile t h e sym bol o f t h e operator L does not necessarily degenerate.T h u s w e h ave th e n otion of ellipticity for th e operator L .T h ese facts in d icate th at the equation 4 = 0 plays an important r o le i n t h e s tu d y o f t h e rig id ity p ro b lem for th e immersion f.O w ing the operator L , h e h a s in d e e d e sta b lish e d a g lo b a l rigid ity theorem ([1 5 ], Theorem 2. 4) w h ic h m a y b e s ta te d as follows : L e t f o b e an immersion M ---> R -w h ich satisfies the following conditions : 1 ) fo is e llip tic , i. e., fo is n o n -d eg en erate and the associated equation 40=0 is elliptic ; 2 ) f o is g lo b ally in fin itesim ally rigid , i. e., e v e ry global solution of th e equatio n 4 0 = 0 is d e riv e d fro m a n infinitesim al Euclidean transformation of R -; 3 ) M is co m p a ct.T h en the theorem states that if tw o imbeddings f and f ': M -+ R " ' lie b o th n ear to f o w it h respect to the 0 -to p o lo g y , a n d i f th e y in d u c e t h e sam e R iem an n ian m etric g, then there is a unique Euclidean transformation a o f R -su c h th a t f '= -- -af.H e has also applied this theorem to the canonical isom etric im bedding f o of a com pact hermitian sym m etric space, M = K/K o , into t h e Euclidean space fr - -R -, f b e in g th e L ie algeb ra o f K , and h as o b tain ed a rigid ity