The Space of Immersions Parallel to a given Immersion
Sheila Carter, Zerrin Şentürk · Journal of the London Mathematical Society · 1994
IMMERSION 405to impose this restriction.For peM, let U be a neighbourhood of p in M such that f\U:U ^ U m+k is an embedding and let v(p) denote the /c-plane which is normal to f(U) at f(p).The total space of the normal bundle N(/) = {(p,V)EMX U m+k :f(p) + ve v(p)} and the endpoint map n:N(f) -* U m+k is denned by n(p,v) =f(p) + v. Let Z c N(/) be the set of singularities of rj.For peM, put N p (/) = N(/) n ({/>} x U m+Ic ) and put S p = S n N p (/).A point xelR m+fc is a focal point of/with base p and multiplicity /I > 0 if the rank of the Jacobian of n at (p, x -f(p)) is m + k -X.The set C(p) of focal points of/with base/?(or centres of curvature), is an algebraic variety in v(p).So C(p) = r,(L p ).Now let U m+k be a parallel normal field.So {(/?,£,{p)):peM) is a parallel section of N(/).Define f ( : M -• U m+k by f^p) =f(p) + £(p)ev(p)-Then/^ is an immersion if and only iff ( (p)$ C{p) for all peM [3].If/ { is an immersion, the index of/ { , indy^, is defined to be the total multiplicity of the focal points of/with base/7 on the line segment between /(/?) and f^p), which is constant over M. In fact the set of focal points of/ with base p is equal to the set of focal points of/^ with base p.We are going to assume that the normal bundle of/has trivial holonomy group.So there exists an orthonormal set of parallel normal fields n x ,..., n k : M m -> R m+k , and a map ¥ : N(/) -> R fc can be defined by ¥(/?, ^f_ x fl c H^/ 7 )) = {a x ,..., a A ).We are going to study R fc \*P( 2).An alternative description of this set in terms of focal points can be obtained as follows.If, for peM, we put ¥ p = *F | N p (/):N p (/) -> (R f c and define ^P^v(/>) -* ^* by • R m+ * be a parallel normal field.Then for some constants a x ,...,a k and for all peM, £(/?) = ^i-i a i n i(P)-Hence we obtain the following result.PROPOSITION 1.1.Let ^ be a parallel normal field on M then for allp,qeM.Alternatively this can be expressed as saying that ¥({(/?, £(/?)) :p e M}) is a single point.Conversely, for any a = (a x , ...,a k )eU k , the normal field £,(a), defined by is parallel and, for all peM, Q> p (f i(a) (p)) = a.So elements a of R k correspond to maps f i{a) : M -* U m+k .Since/ {(a) is an immersion if and only iff i{a) (p) $ C(p) for all p e M, we obtain the next result.