On the Components of the Push-out Space with Certain Indices

On the Components of the Push-out Space with Certain Indices

YUSUFKAYA

ABSTRACT- Given an immersion of a connected,m-dimensional manifoldMwithout boundary into the Euclidean (m‡k)-dimensional space, the idea of thepush- out space of the immersion under the assumption that immersion has flat normal bundle is introduced in [3]. It is known that thepush-out spacehas finitely many path-connected components and each path-connected component can be assigned an integer called the index of the component. In this study, whenMis compact, we give some new results on thepush-out space. Especially it is proved that if thepush-out spacehas a component with index1;then the Euler number ofM is0and if the immersion has a co-dimension2, then the number of path-connected components of thepush-out space with index (m 1) is at most 2.

1. Introduction

Throughout we assume M (or M^m) is an m-dimensional connected smooth (C¹) manifold without boundary. The tangent space ofMat a point pwill be denoted byT_pM.

f :M^m!R^m‡k will be assumed a smooth immersion or embedding into Euclideanm‡kspace, i.e. f hasco-dimension k. In this case

dfp:TpM^m!Tf(p)(R^m‡k)ˆ ff(p)g R^m‡k R^m‡k

is an injection. We identifyT_pMwithIm df_p,8p2M:In this way, we can assume thatfis an isometric immersion. There is a standard inner product h;ionR^m‡k. So we can define the normal space atpas the normal com- plement ofIm dfp. Letnp(f) denote thek plane which is normal tof(M) at

(*) Indirizzo dell'A.: BuÈlent Ecevit University, Faculty of Arts and Sciences, Department of Mathematics, 67100, Zonguldak/Turkey.

E-mail: yusufkaya69@gmail.com

f(p). The total space of the normal bundle is defined by N(f)ˆ f(p;v)2MR^m‡k:f(p)‡v2n_p(f)g:

Note thatN(f) is an (m‡k)-dimensional smooth manifold.

A normal field on M for f is a smooth map j:M^m!R^m‡k where f(p)‡j(p)2n_p(f) for allp2M.

With this notation, the endpoint mapE:N(f)!R^m‡k is defined by E(p;v)ˆf(p)‡v, andEis known to be a smooth map.

1.1 ±Immersions of manifolds and focal points

DEFINITION1. A point x2R^m‡kis a focal point of f(M)with base p if E is singular at(p;x f(p)), i.e.(p;x f(p))is a critical point of E. The focal point has multiplicitym>0 if rank(Jacobian E)ˆm‡k m .

The set of focal points of f (or f(M) ) with basepwill be denoted by F_p(f):This is an algebraic variety, that is, it is a set of zeros of a poly- nomial with degree at most m in k variables in n_p(f) and in general it can be quite complicated [8]. In this study, we will be considering the simplest case. We remark that by [6], x2F_p(f) iff x2n_p(f) and xˆf(p)‡1

lj(p) where j(p)ˆ x f(p) x f(p)

k k and l is an eigenvalue of the shape operatorAj(p):TpM !TpM, i.e.lis a principal curvature of f at

f(p) in the normal direction j(p):

Forx2R^m‡k thedistance functionfor f,L_x:M^m!Ris defined by L_x(p)ˆ kx f(p)k². Using [6], the pointp2Mis a critical point ofL_xif and only ifx2n_p(f) and furtherpis a non-degenerate critical point ofL_xif and only ifxis not a focal point of f with basep. So,

F_p(f)ˆ fx2R^m‡k :pis a degenerate critical point of L_xg:

We use this characterisation ofF_p(f) to calculate focal points with basep.

Further, using [6] again, the index ofL_xat a non-degenerate critical point p2Misequal to thenumberoffocal pointsoffwithbasepwhich lieontheline segment fromf(p) tox, each focal point being counted with its multiplicity.

1.2 ±Parallel immersions to a given immersion

DEFINITION 2. Let f :M^m!R^m‡k be an immersion and fh₁;h₂;. . .;h_kgbe an orthonormal set of normal fields for f in a neigh-

bourhood of some point p2M. A normal field j for f is said to be a parallel normal field; if @j

@p_i;h_j

ˆ0 for all p2M; where iˆ1;. . .;m; jˆ1;. . .;k and p₁;. . .;p_m is a coordinate system in a neighbourhood of p2M.

Since we assumeMis connected, note that a parallel normal field onM has constant length.

Let f :M^m!R^m‡kbe an immersion and assumej:M^m!R^m‡kis a parallel normal field for f . The map fj:M^m!R^m‡k is defined by

f_j(p)ˆf(p)‡j(p):

Iff_jis an immersion, it is called aparallel immersiontofandjis said to beimmersive. We remark that, for allp2M, the normal planes of f and f_jat eachp2M are the same.

If f_jis an immersion, then the indexof f_j,ind f_j, is defined to be the total multiplicity of the focal points of f with basepon the line segment between f(p) and fj(p), this index is shown to be constant overM by the following well-known fact. We call this number as the index of the im- mersive parallel normal fieldjas well.

LEMMA 1. Let f :M^m!R^m‡k be an immersion and let j:M^m !R^m‡k be a parallel normal field for f . Then the following are satisfied.

(i) fj is an immersion if and only if for all p2M; fj(p)is not a focal point of f with base p

(ii) x2R^m‡kis a focal point of f_jwith base p if and only if x is a focal point of f with base p. So, F_p(f_j)ˆF_p(f)for all p2M.

1.3 ±The push-out space of immersions with flat normal bundle

LetMbe a connected,m-dimensional manifold andf :M^m!R^m‡kbe an immersion. If for allp2M;there exists a neighbourhoodUMofp and a parallel normal frame field for f on U, then it is said that f has locally flat normal bundle. The normal bundleN(f) is flat (or globally flat) if there exists a global parallel normal frame onM.

If the immersion f has locally flat normal bundle, then at each base

point p2M, the focal set onnp(f) is a union of at mostm hyperplanes (which is the simplest set that can occur as focal set, if non empty) where each plane is counted with its proper multiplicity [8, pp. 69-70]. A gen- eralisation and the converse of this result can be derived from [4].

First, we assume that the normal bundle of f is globally flat. So there exists an orthonormal set of parallel normal fieldsj₁;. . .;j_k :M^m!R^m‡k for f: For each p2M, a map W_p:np(f)!R^k can be defined by W_p

f(p)‡P^k

iˆ1a_ij_i(p)

ˆ(a₁;. . .;a_k). For each p2M; we denote V_pˆ R^knW_p(F_p(f)). Then, the push-out spaceof the immersion f is defined by

V(f)ˆ \

p2M

V_p:

This set is essentially defined and many properties of it are studied in [3].

For example,V(f) has finitely many path-connected components with each component convex and each component can be assigned an integer called as index. Further, ifMis compact, each component is open. The definition of V(f) depends on the choice ofj₁;. . .;j_k, but, it is shown in [3] that different choices produces an isometric set. We are going to study some properties of V(f) which are related to number of path-connected components ofV(f) with certain indices and some relations with the Euler characteristic ofM(when Mis compact).

As pointed out in [3] we can next consider an immersion f of m -di- mensional manifoldMwhich has locally flat normal bundle but the normal holonomy group is nontrivial. In this case we can take the simply connected covering spaceM~ ofMwith covering mapp: ~M^m!M^mand work with the immersion~f ˆfp: ~M^m!R^m‡k which has globally flat normal bundle with trivial normal holonomy. We know that f and~f have the same focal set:

PROPOSITION1. With the notation above, F_p(f)ˆF~p(~f)for all p2M and~p2M with~ p(~p)ˆp;wherep: ~M !M is the covering map.

PROOF. Letx2R^m‡kand defineL~x: ~M^m!R(distance function for the immersion~f) by

L~x(~p)ˆ kx ~f(~p)k²ˆLxp(~p);

whereL_x:M^m!Ris the usual distance function for f. Sincepis an im-

mersion, ~pis a degenerate critical point ofL~x if and only ifp(~p) is a de- generate critical point ofLx. ThereforeFp(f)ˆF~p(~f) for all ~p2M~ and

p2Mwithp(~p)ˆp: p

So this result allows V(f) to be defined by V(f)ˆV(f p)ˆV(~f).

This is useful especially when we have an immersion of a nonorientable manifold with locally flat normal bundle where obviously the normal holonomy group is nontrivial. Consequently, by replacing f with ~f if necessary, we may assume that f has globally flat normal bundle with trivial normal holonomy group. Remark that M~ may fail to be compact again even M is compact, but we can use critical point theory of distance function thorough the immersion of Mto deduce some results on V(~f).

Let aˆ(a₁;a₂;. . .;a_k)2V(f). As in [3], definej(a):M^m!R^m‡k by j(a)(p)ˆP^k

iˆ1a_ij_i(p);wherej₁;j₂;. . .;j_k are unit parallel normal fields on Mforming a basis for the normalk-plane atf(p) for allp2M. Then it is easy to check that j(a) is an immersive parallel normal field for f onM.

With this notationV(f) can be defined as V(f)ˆ

a2R^k:f(p)‡j(a)(p) is not a focal point of f with basep;8p2M : DEFINITION3. Let a2V(f). The index of a, ind a, is defined to be the index of the immersion f_j(a).

We know by [3] that ifAis a path-connected component ofV(f) and if a;b2A;thenind aˆind b. Then theindexofAis defined to beind afor somea2Awhich is constant overA. So each path-connected component of V(f) can be assigned a number, called its index. We will denote the union of the path-connected components ofV(f) with indexm byV^m. So V(f)ˆ V⁰[V¹[ [V^m: Note that V⁰ is always non empty and the others may be empty or not.

2. Path-connected components and their respective indices.

In this section, firstly, we give an example to illustrate theV(f) for a given embedding f with flat normal bundle and then we prove some general results onV(f).

EXAMPLE1. Let~f :RR!S³R^2‡2 be given by

~f(u;f)ˆ 1

p2(cosu;sinu;cosf;sinf);

then ~f induces an embedding f of S¹S¹ into S³R^2‡2 by taking u mod 2p,fmod 2pand alsoV(f)ˆV(~f). Now,

j₁(u;f)ˆ~f(u;f)ˆ 1

p2(cosu;sinu;cosf;sinf); j₂(u;f)ˆ 1

p2( cosu; sinu;cosf;sinf)

are unit parallel normal fields to~fand form a basis for the normal planes for all (u;f)2RR. Putj(u;f)ˆtj₁(u;f)‡sj₂(u;f);for somet;s2R;then

~fj(u;f)ˆ~f(u;f)‡tj1(u;f)‡sj2(u;f):

ˆ 1

p2…(1‡t s)cosu; (1‡t s)sinu;(1‡t‡s)cosf;(1‡t‡s)sinf†:

Using the distance functionLx(u;f)ˆ kx ~f(u;f)k² forxˆ(x1;x2;x3;x4)2 R⁴, we get

@Lx

@u ˆ 2

p2(x1sinu x2cosu); @Lx

@f ˆ 2

p2(x3sinf x4cosf);

@²Lx

@u² ˆ 2

p2(x1cosu‡x2sinu); @²Lx

@f² ˆ 2

p2(x3cosu‡x4sinu);

@²Lx

@f@u ˆ @²Lx

@u@fˆ0:

Then

Hess L… † ˆx Hˆ

@²Lx

@u² 0

0 @²Lx

@f² 2

66 64

3 77 75:

So~f_j(u;f) is a focal point of~fat (u;f)()(u;f) is a degenerate critical point ofL_x. From the equations@L_x

@u ˆ0ˆ@L_x

@f, we obtain, for each (u;f)2R², xˆ~f_j(u;f) for somet;s2R. By replacingxby~f_j(u;f) and usingdet Hˆ0

we get

det 2

p2(1‡t s) 0

0 2

p2(1‡t‡s) 2

66 4

3 77

5ˆ0()(1‡t‡s)(1‡t s)ˆ0 ()(1‡t)² s²ˆ0

()sˆ (1‡t):

Therefore the focal set of~f with base (u;f)2RRis a pair of lines per- pendicular to one another which is the same for all base points (u;f).

Consequently

V(~f)ˆV(u;f)(~f)ˆV_(u⁰_;f⁰₎(~f); 8(u;f);(u⁰;f⁰)2RR:

Then,V(~f) has four path-connected components since eachV_(u;f)(~f) has four path-connected components; one of index 0, two of index 1, and one of index 2. Hence the same is true for V(f); as V(f)ˆV(~f), see the Figure 1. Then V(f)ˆV⁰[V¹[V² and in the Figure 1, we put V¹ˆA[B.

Figure 1

We know by [3],V(f) can have at most one component with indexmand if there is such a component, it is unbounded. Note that, we have examples of immersions such thatV⁰is bounded.

THEOREM1. Let f :M^m!R^m‡k be an immersion of a compact, m- dimensional manifold such that f has flat normal bundle. IfV(f)has a component with index m, thenV⁰is unbounded.

PROOF. Leta2V^mand take the immersive parallel normal fieldj(a) . Then W_p¹(a)ˆf_j(a)(p);8p2M and index f_j(a)ˆm. So for allp2M, the total multiplicity of focal points with basepon the line segment fromf(p) to

f_j(a)(p) ism. Therefore there are no focal points on the rays Rpˆf(p)‡tj(a)(p)2np(f): t1 Q_pˆf(p)‡tj(a)(p)2n_p(f): t0

: Also8p2M; W_p(Q_p)ˆ fta:t0g V_pand so

fta:t0g \

fV_p:p2Mg ˆV(f):

Hencefta:t0g V⁰andV⁰is unbounded. p

PROPOSITION2. Let f :M^m!R^m‡kbe an immersion with flat normal bundle and assumej;h:M^m!R^m‡kare immersive parallel normal fields for f with indiceslandm respectively. Then, the number of focal points with base p on the line segment from fj(p)to fh(p)is constant for all p2M and it isl‡m 2l for some l2Nwheremaxf0;l‡m mg lminfl;mg:

PROOF. As in Lemma 4.4 of [3], since f_j‡(h j)ˆf_h, soh j is an immersive parallel normal field for the immersionf_jand we need to find its index for fjwhich is constant. Here we try to formulate this constant.

Let p2M and put xˆfj(p) and yˆfh(p). If f(p);x;y are collinear, then the number of focal points with basepon the line segment fromf_j(p) to f_h(p) is l‡m orjl mjwith respect to positioning of f(p) and we can take lˆ0 or lˆl or lˆm : Otherwise, take the triangle on n_p(f) with vertices f(p);x;y, and consider the 2 -plane sayQ(p) which contains this triangle. We know thatQ(p)\Fp(f) is a union of at mostmlines if it is non empty, sinceF_p(f) is a union of at mostmhyperplanes onn_p(f) [8].

Ifu;v2n_p(f), then the notationuvdenotes the line segment fromuto v. We know the total multiplicity of focal points onf(p)xisland onf(p)yis m. Now, let l(p)0 be an integer and assume that l(p) lines(counting multiplicities) meet both of the edges f(p)x and f(p)y. Clearly 0l(p)minfl;mg. Then the remainingl l(p) lines intersectingf(p)x must intersectxy. And similarly the remainingm l(p) lines intersecting f(p)y must intersectxy. So we get the total multiplicity on xyis exactly

l l(p)‡m l(p)ˆl‡m 2l(p). So we deduce the total multiplicity of focal points on the line segment fromfj(p) tofh(p) isl‡m 2l(p). But the index of the parallel immersion… †f_{j (h j)}tof_jis a constant number, sol(p) is constant for allp2M.

Put lˆl(p). Since there exist at most m lines on Q(p), then l‡m ml. Then maxf0;l‡m mg lminfl;mg. p COROLLARY1. Iflˆmˆ1in Proposition2, then, for allp2M;the number of focal points with basepon the line segment fromf_j(p)tof_h(p) is2(wherelˆ0).

THEOREM2. Let f :M^m!R^m‡k be an immersion with flat normal bundle and assumej:M^m!R^m‡kis an immersive parallel normal field for f .

(i) There exists a w2R^k such that V(fj)ˆV(f) w, where V(f) wˆ fa w:a2V(f)g,

(ii) if A is a path-connected component ofV(f)with indexm and if the index of f_jisl, then there exists an l2Nsuch that A w is a path- connected component of V(f_j) with index l‡m 2l, where maxf0;

l‡m mg lminfl;mg.

PROOF. (i) Since f has flat normal bundle, there exists a set of ortho- normal parallel normal fieldsfj₁;j₂; ;j_kgforming a basis of the normal space at each base pointp2M. We will use this basis to defineV(f) and V(fj). We can write

jˆw1j₁‡ ‡wkj_k

for some constants w₁;. . .;w_k2R and put wˆ(w₁;. . .;w_k)2R^k. Then we can easily see that a2V(f)()a w2V(f_j). In fact, let aˆ(a₁;. . .;a_k)2V(f). We knowF_p(f)ˆF_p(f_j) for allp2Mby Lemma 1 (ii). Then, for allp2M

f(p)‡a₁j₁‡ ‡a_kj_k2= F_p(f)()f ‡j‡a₁j₁‡ ‡a_kj_k j

ˆf_j‡(a1 w1)j₁‡ ‡(a_k w_k)j_k2= Fp(f_j):

Soa w2V(f_j) and thereforeV(f_j)ˆV(f) w.

(ii) Leta2A, then clearlya w2V(f_j) by Theorem 2 (i), henceA w is a path-connected component of V(f_j). Since A is a path-connected component ofV(f) with indexm, there exists an immersive parallel normal field h for f with index m and W_p…f(p)‡h(p)† ˆa for all p2M. As in

Proposition 2, fj‡(h j)ˆfh, so h j is an immersive parallel normal field for fj and its index for fj is l‡m 2l for some l2N where

maxf0;l‡m mg lminfl;mg. p

The following result concerns the positioning of the path-connected components ofV(f) inR^k.

THEOREM3. Let f :M^m!R^m‡k be an immersion with flat normal bundle. Let A;B be path-connected components ofV(f)with indexlandm respectively. Ifl‡m>m, then there exists a hyperplane inR^ksuch that A and B lie on one side of the hyperplane andV⁰lies on the opposite side of the hyperplane.

PROOF. LetA;Bbe path-connected components ofV(f) with indexl andmrespectively anda2A,b2B. Then there exist immersive parallel normal fieldsj;hfor f such thatindex f_jˆl,index f_hˆm and also for all p2M; W_p¹(a)ˆfj(p) and W_p¹(b)ˆfh(p). Now consider the normal plane np(f) for a fixed p2M and the focal hyperplanes P1;. . .;Ps on np(f) with their respective multiplicity w_i where 1is, sm and w₁‡ ‡w_sm.

Since f_jhas indexl, the line segment joining f(p) tof_j(p) must cross Pa1;. . .;Pal where ll, wa1‡ ‡wal ˆl and similarly the line seg- ment joining f(p) to fh(p) must cross Pb₁;. . .;Pb_d where dm, w_b1‡ ‡w_bd ˆm.

Here,P_a1;. . .;P_al;P_b1;. . .;P_bdare not all distinct sincel‡m>m. So letP2 fP_a1;. . .;P_alg \ fP_b1;. . .;P_bdg. Then we claim thatA;Bstay on one side of the hyperplaneW_p(P)ˆLin R^k. SetW_p(Pi)ˆLi, 1is . Since eachLidividesR^kinto two open connected regions, we identify them by writingL_i for the region including the origin andL^‡_i for the other part.

Then, V⁰L_i for all 1is, AL^‡_a1\ \L^‡_al and B L^‡_b1\ \L^‡_bd. ThereforeAandBstay inL^‡, and henceLis the hyper-

plane we are seeking. p

3. Number of path-connected components ofV(f)with certain indices It is interesting to know the number of path-connected components of V(f) with their respective indices for an immersion f ofM as it includes some information on the geometry and the topology of them-dimensional compact manifold M. Here, we prove that if we have a path-connected

component of V(f) with index 1, then the Euler characteristic of Mis 0.

Secondly, we prove that the number of path-connected components ofV(f) with index (m 1) is at most 2 for a co-dimension 2 immersion.

THEOREM4. Let f :M^m !R^m‡kbe an immersion of compact mani- fold with flat normal bundle and letx(M)6ˆ0where m is an even number.

ThenV¹ˆ ;.

PROOF. IfV¹6ˆ ;, then there exists a unit parallel normal fieldjforf such that fsjˆf ‡sj is an immersion with index 1 for some s>0. So 8p2M;there exists only one focal pointc(p) of multiplicity 1 on the line segment from f(p) to f_sj(p) such that c:M^m !R^m‡k; p!c(p) is con- tinuous. Definel:M^m !Rby

l(p)ˆ 1 kf(p) c(p)k:

Thenlis continuous as it is the principal curvature function offin the unit normal directionj. Alsolis smooth since it is of constant multiplicity 1 onM [7]. So the principal direction corresponding to the principal curvaturel(p) defines a nonzero smooth tangent vector field onMwhich has no zeros. So considering thatM is compact,x(M)ˆ0 by the PoincareÂ-Hopf Theorem in [5]. But this contradictsx(M)6ˆ0. ThereforeV¹ˆ ;. p A generalisation of this theorem to any odd indexed component is proved in [1] by a different method. Present method here may not be generalized, because respective vector field can fail to be smooth.

DEFINITION4. Let f :M^m !R^m‡k be an immersion with flat normal bundle, then d(f)is defined to be the total number of the path-connected components ofV(f).

It was proved in [3] thatd(f)a(m;k) wherea(m;k) is the number of path-connected regions in the complement of m hyperplanes in general position inR^kas

a(m;k)ˆ

2^m ifm k

X^k

iˆ0

m

i ifm>k 8>

><

>>

: :

COROLLARY 2. Let f :M² !R^2‡k be an immersion with flat (or locally flat) normal bundle of a compact surface for some k1 and let x(M)6ˆ0. ThenV¹ ˆ ;and sod(f)2.

PROOF. By Theorem 4,V¹ˆ ;and alsoV⁰,V²are connected [3], hence d(f)2. Of courseV²can occur, definitely when f is spherical, [2]. p EXAMPLE2. For the homology groups of real projective spaceRP^m, we know thatH_i(RP^m;Z₂)ˆZ₂ for alliˆ1;2;. . .;m. Then

x(RP^m)ˆ 0 ; if mis odd 1 ; if mis even (

:

So, by Theorem 4, if f :RP^m !R^m‡k is any immersion with locally flat normal bundle, we haveV¹ˆ ;formis even.

Let f :M² !R^2‡k be an immersion of a 2 -dimensional manifold M with flat normal bundle. Then for any pointp2M,F_p(f) is a union of at most 2 hyperplanes inn_p(f). SoF_p(f) can dividen_p(f) into at most 4 path- connected regions, and the number of path-connected components ofV(f) with index 1 can be at most 2 for anyk2.

In the following theorems we generalize this and prove a result con- cerning the number of path-connected components of V(f) with index (m 1) wherem3.

THEOREM5. Let m2 and f :M^m !R^m‡k be an immersion with flat normal bundle. Assume A;B are two different path-connected com- ponents ofV(f)both with index(m 1)and a2A, b2B. Then for each p2M, all the focal hyperplanes inn_p(f)meet the triangle4with vertices f(p),W_p¹(a),W_p¹(b), and moreover the total number of focal points on the line segment fromW_p¹(a)toW_p¹(b)is exactly2.

PROOF. Let a2A, b2B. Then there are corresponding parallel normal fields j ˆj(a) and h ˆj(b) say, such that index fjˆindex fhˆm 1. By Proposition 2, for allp2M, we have total number of focal points between f_j(p)ˆW_p¹(a) andfh(p)ˆW_p¹(b) is 2…m 1† 2lfor some l2Nwherem 2lm 1. Sinceaandbare in different components, there must be at least one focal point betweenf_j(p) andf_h(p) for allp2M:

Solˆm 2:

LetQnp(f) be the plane including the triangle4with verticesf(p), fj(p),fh(p). Sincelˆm 2, we have proved that the total multiplicity of focal points on f_j(p)fh(p) is exactly 2 for all p2M and hence there are exactlymfocal lines meeting with the triangle4as required. Since there aremlines inQ, this implies that all focal hyperplanes onn_p(f) meet with

the triangle4, for allp2M: p

THEOREM 6. Let f :M^m !R^m‡2 be an immersion of a compact manifold such that f has flat normal bundle, where m3. Then the number of path-connected components ofV(f)with index(m 1)is at most2.

PROOF. Assume there exist at least three path-connected components ofV(f) with index (m 1), sayA;B;C. Takea2A,b2B,c2C. Letp2M be an arbitrary point and consider the points xˆW_p¹(a), yˆW_p¹(b), zˆW_p¹(c) onn_p(f). Clearlyx;y;zare nonfocal distinct points, sincea;b;c are in different components.

Sincea;b;care in different components there is at least one focal point on each line segmentxy;yz;zx. So we can check that the pointsx;y;f(p) cannot be collinear. Assume they lie on a line`say. If f(p) is onxythen the total multiplicity of focal points on`is at least (2m 2) which is not pos- sible for m3 , since 2m 2>m. If f(p) is not onxy we get the total multiplicity of focal points onf(p)xorf(p)yis at leastmdepending on the positioning off(p) on`with respect to the pointsx;y. This contradicts the hypothesis that this number is (m 1) . By a similar discussion we get the pointsx;z;f(p) ory;z;f(p) orx;y;z;f(p) cannot be collinear.

By Theorem 5, all of the focal lines must meet the triangle with vertices x;y;f(p) and further the total multiplicity of focal points onxyis exactly 2.

Similarly we get the same result considering the triangles with vertices y;z;f(p) andx;z;f(p).

We next showx;y;zare not collinear. For ifx;y;zall lie on a line then by the above argument the total multiplicity of focal points on each line segmentxy;yz;zxis exactly 2. Without loss of generality we can assumey is on xz. Then we obtain the total multiplicity of focal points on xz is 2‡2ˆ4 which is a contradiction.

Now consider the triangle with verticesx;y;z. There are 3 cases to be considered.

CASE1. Assume f(p) is in the region I bounded by the triangle with verticesx;y;zas shown in Figure 2. By Theorem 5 there exists at least one focal line meeting withxf(p) andzf(p) considering the triangle with ver-

ticesx;f(p);z. Similarly there exists one focal line meeting withxf(p) and yf(p) considering the triangle with verticesx;f(p);y. And also there exists one focal line meeting withyf(p) andzf(p) considering the triangle with vertices y;f(p);z. These focal lines are necessarily all different and to- gether bound f(p). This implies that f(p) is in a bounded region of the complement of the focal lines onn_p(f).

CASE2. Assumef(p) is in the region II as shown in Figure 2. Consider the triangle with verticesz;f(p);y. By Theorem 5, there must be a focal line meeting withf(p)zandzyand this line must necessarily meetxf(p) andxy. Similarly by considering the triangle with verticesx;f(p);z, there must be a focal line meeting with f(p)z and zx and this line must ne- cessarily meetf(p)yandxy. Now we get at least 2 focal points onxy. But again by Theorem 5 and considering the triangle with verticesx;f(p);y, it is exactly 2. So there are no more focal lines meeting with xy. So far we have one focal line meeting bothxf(p) andzf(p). By Theorem 5 and con- sidering the triangle with vertices x;f(p);z, we need (m 3) more focal lines meeting withxf(p) andzf(p) which must necessarily meet withyf(p).

And one more focal line meeting bothxf(p) andxzwhich must necessarily meet withxyorzy. We know there are no more focal lines meeting withxy.

So the focal line meeting bothxf(p) andxzmust necessarily meet withzy.

This implies that for allp2M,zˆfj(a)(p) is bounded by focal lines onnp(f) where the immersive parallel normal fieldj(a) is corresponding toa.

CASE3. Now assume f(p) is in the region III as shown in Figure 2.

Then we know every focal line must meet the triangle with vertices f(p);x;y. But there must be a focal line meeting with f(p)z and zx si- multaneously. So this line cannot meet the triangle with verticesf(p);x;y.

This gives a contradiction by Theorem 5. So we deduce that Case 3 cannot occur.

Sincepis an arbitrary point inMandW_p ¹ is an isometry, then either Case 1 holds for allp2Mor Case 2 holds for allp2Mi.e. either f(p) is bounded by focal lines onnp(f) orfj(a)(p) is bounded by focal lines onnp(f) for allp2M.

Now, for somew2R^m‡2;take the distance functionL_wforf. SinceMis compact, there is a critical point ofL_w with indexm. So the total number of focal points with basepon the line segment fromwtof(p) ismand so there is no focal point with basepon the rayff(p)‡t w f… (p)† jt0g np(f):

This implies that for somep2M;f(p) is not bounded by focal hyperplanes on np(f) and a similar statement is true for f_j(a)(q) considering the immersion

fj(a)for someq2M. So there cannot be such path-connected components A;B;CofV(f). ThereforeV(f) can have at most two path-connected com- ponents with index (m 1).

Figure 2 p

REMARK1. Theorem 6 is not true for an immersion with co-dimension k>2. We can see that by taking product immersions. In Example 1, we have an embedding f of T² into S³R⁴ such that V(f) has two path- connected components with index 1:Now take

ff :T²T² !R^4‡4

by (f f)(p;q)ˆ…f(p);f(q)†wherep;q2T². Note that, by Theorem 4.2 of [3],ffhas flat normal bundle andV(ff)ˆV(f)V(f), sincefhas flat normal bundle. Then, we can easily check that V(ff) has 4 path- connected components with index 3.

Consequently, form>2 andk>2, it is a considerable question to ask what is the maximum number of path-connected components of V(f) with index…m 1†. This might be at mostk.

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Manoscritto pervenuto in redazione il 24 maggio 2010.