Available at: off IC/2001/96 United Nations Educational Scientific and Cultural Organization and International Atomic

Transcription

1 Available at: off IC/2001/96 United Nations Educational Scientic and Cultural Organization and International Atomic Energy Agency THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS ISOMETRIC C 1 -IMMERSIONS FOR PAIRS OF RIEMANNIAN METRICS Giuseppina D'Ambra 1 Department of Mathematics, University of Cagliari, Cagliari, Italy and Mahuya Datta 2 Department of Mathematics, University of Calcutta, Calcutta, India. Abstract Let h 1 ;h 2 be two Euclidean metrics on R q, and let V be a C 1 -manifold endowed with two Riemannian metrics g 1 and g 2. We study the existence of C 1 -immersions f : (V;g 1 ;g 2 )! (R q ;h 1 ;h 2 ) such that f Λ (h i )=g i for i =1; 2. MIRAMARE TRIESTE August dambra@vaxca1.unica.it 2 mahuyad@hotmail.com

2 1. Introduction In 1954 Nash [4] proved his C 1 -isometric Embedding Theorem saying that if an n-dimensional Riemannian manifold (V;g) admits a C 1 -immersion (resp. embedding) in R q, where q dim V + 2, then it admits a C 1 -isometric immersion (resp. embedding) in R q. Kuiper [3] improved this result by showing that it is true even when q dim V +1. The key idea in the proof is to start with a g-short C 1 map and to approach g through a sequence of g-short C 1 -maps improving the approximation in every successive step and keeping control over the rst derivative of the map so that nally the required isometric C 1 -map comes as the C 1 -limit of the sequence. The problem of C 1 -isometric immersions in a general Riemannian Manifold has been treated by Gromov through the Convex Integration Theory [2] which accommodates the essence of Nash-Kuiper techniques. The aim of this paper is to generalize the result of Nash to the case where the manifolds come with pairs of Riemannian metrics. The problem we consider can be formulated as follows: Let h 1 be the standard Euclidean metric on R q with canonical coordinates (x 1 ;x 2 ;:::;x q ) and h 2 = P q i=1 2 i dx2 i, where we assume that 0 < 1 < 2 < < q. Consider a C 1 -manifold V with two C 1 -Riemannian metrics g 1 and g 2. The problem is to determine when there exists a C 1 -immersion f :(V;g 1 ;g 2 )! (R q ;h 1 ;h 2 ) such that f Λ (h 1 )=g 1 and f Λ (h 2 )=g 2 : It is evident that not all pairs (g 1 ;g 2 ) can be induced from (h 1 ;h 2 ) by a map. In fact, (g 1 ;g 2 ) must satisfy the inequality 2 1 g 1» g 2» 2 q g 1. The above condition in general is not sufcient. For example we consider the linear situation, when V = R n, g 1 = P n i=1 dy2 i, and g 2 = P q i=1 μ2 i dy2 i with μ i > 0 for all i =1;:::;n. It may be proved by simple calculation that if there are 2n indices 1 ;i 2 ;:::;i 2n g such that i2k 1 <μ k < i2k for k = 1; 2;:::;n, then we obtain an injective linear map R n! R q inducing (g 1 ;g 2 ) from (h 1 ;h 2 ). In particular, let a = min k μ k and b = max k μ k be two positive numbers. If there are n many i 's smaller than a and an equal number of i 's greater than b, then there exists an injective linear map which is isometric with respect to both the pairs (g 1 ;h 1 ) and (g 2 ;h 2 ). On the other hand if we take g 1 = g 2 then the existence of such a linear map necessarily implies the existence of an injective linear map L : R n! R q which is(h 2 h 1 )-isotropic, that is L Λ (h 2 h 1 )=0. If q + (h) and q (h) respectively denote the positive and the negative ranks of h = h 2 h 1 then such an isotropic map exists only if q + n and q n, for the set of h-isotropic vectors contains at the most an m-dimensional subspace for m = min(q + ;q ). We now state the main result of this paper and an immediate corollary to it. 2

3 Theorem 1.1. Let (R q ;h 1 ;h 2 ) be asabove and let V be ann-dimensional C 1 -manifold. Let a; b be two positive numbers, a<b, such that i <afor all i» 3n +2 and i >bfor i q 3n 1, and none of the i 's lies between a and b. If g 1 and g 2 are two Riemannian metrics on V satisfying the relation a 2 g 1 < g 2 < b 2 g 1 then there exists a C 1 -immersion f : V! R q such that f Λ (h 1 )=g 1 and f Λ (h 2 )=g 2 : Corollary 1.2. Let (V;g) be a smooth Riemannian manifold of dimension n, and let (R q ;h 1 ;h 2 ) be as in Theorem 1.1. If h 1 h 2 is non-singular, and if the positive rank q + and the negative rank q of h 1 h 2 satisfy q + 3n +2 and q 3n +2 then there exists a C 1 -immersion f : V! R q such that f Λ (h 1 )=g = f Λ (h 2 ). Observe that the map f in the corollary is, in particular, an isotropic immersion relative to h 1 h 2. One may compare the above corollary with a result of Gromov ([2], 2.4.9(B)) whichsays that if (W;h) is a pseudo-riemannian manifold with q + ;q 2n then an arbitrary C 1 -manifold V admits a isotropic C 1 -immersion in W. We shall closely follow Nash in our proof of Theorem 1.1. We explain the basic ideas of Nash's techniques [4] in Section 2. Through Section 3 to Section 5 we shall make preparation for the adaptation of Nash's techniques in our context. Finally in Section 6 and 7 we prove Theorem Preliminaries Here we briefly discuss Nash's proof of C 1 -Isometric Embedding Theorem. For a detailed treatment we refer the reader to [1] [4]. Theorem 2.1. (Nash) If a Riemannian manifold (V;g) admits a smooth immersion (or embedding) in R q, then there exists an isometric C 1 -immersion (respectively, a C 1 -embedding) f :(V;g)! (R q ;h), provided dim V» q 2, where h denotes the canonical Euclidean metric on R q. To prove the theorem Nash starts with a C 1 -immersion f 0 : V! (R q ;h) which is strictly short relative to the C 1 -Riemannian metric g on V. Recall that a C 1 -map f 0 : V! R q strictly short if the difference g 1 f Λ 0 h 1 is positive denite. This is expressed by the notation g 1 f Λ 0 h 1 > 0org 1 >f Λ 0 h 1. Short immersions V! R q exist whenever V admits an immersion in R q. The basic idea is to construct a sequence of strictly short C 1 -immersions f j :(V;g)! (R q ;h) satisfying the two properties: 1. kg f Λ j (h)k 0 < 2 3 kg f Λ j 1 (h)k 0; 2. kf j f j 1 k 1 <c(n)kg f Λ j 1 (h)k 1 2 0, where k:k r denotes the norm in the C r topology and c(n) is a constant depending on the dimension n of the manifold V. It then follows that the sequence ff j g is a Cauchy sequence in 3 is

4 the C 1 -topology and hence it converges to some C 1 -function f : V! R q such that f Λ (h) =g. The problem therefore boils down to the following: Lemma 2.2. Assume q dim V +2 and let f 0 : V! (R q ;h) be a strictly short C 1 -immersion. Then there exists a C 1 -immersion f 1 : V! (R q ;h) such that: (i) f 1 is strictly short, (ii) kg f Λ 1 (h)k 0 < 2 3 kg f Λ 0 (h)k 0; (iii) kf 1 f 0 k 1 <c(n)kg f Λ 0 (h)k1=2 0. Proof. The construction of f 1 involves a sequence of successive `stretching' and `twisting' on f 0. We x a locally nite covering U = fu p : p 2 Zg of V by contractible coordinate neighbourhoods U p in V. Nash noted in his paper that a positive denite C 1 -metric g on a manifold V admits a decomposition as stated below: (1) g = 1X i=1 f 2 i dψ 2 i ; where f i and ψ i are C 1 -functions on V with support of f i compact and contained in some U p 2U. Further, almost all f i 's vanish on a U p, and there are at most N many f i 's which are simultaneously non-zero at any point of V, where N is a constant depending on the dimension of V. This observation plays a crucial role in his proof of Theorem 2.1. Since f 0 is strictly short, f = g f Λ 0 (h)is positive denite. Hence f=2 can be decomposed as f=2 =P 1 i=1 f2 i dψ2 i, where ψ i and f i are C 1 -functions on V as in equation (1). We rst construct a sequence of C 1 -immersions, f μ f i g with μ f 0 = f 0. In the i-th successive step we deform μ f i 1 on some U p 2 U where U p ff supp f i, to obtain μ f i such that the induced metric is approximately augmented by the factor f 2 i dψ2 i. Fix any " 1 > 0. We describe how f 0 can be deformed within its " 1 -neighbourhood (relative to the C 0 topology) to a C 1 -immersion μ f 1 such that the induced metric μ f Λ 1 (h) =f Λ 0 (h)+f2 1 dψ2 1 + O(" 1 ). Normal Extension(stretching): From the above decomposition of f=2 we obtain an U p 2 U such that supp f 1 ρ U p. We extend the map f 0 to ~ f 0 : U p R 2! R q in such away that ~f Λ 0 (h) =f Λ 0 (h) Φ dt 2 Φ ds 2 on TR q j Up, where (t; s) denote the global coordinates on R 2. To get the extension we merely have tochoose two vector elds namely 1 and 0 1 on U p which are mutually orthogonal and also normal to TV. Since f 0 is an immersion and q dim V +2 such vector elds always exist on contractible sets. Then we dene ~ f 0 : U p R 2! R q by the formula: (y; t; s) 7! f 0 (y)+t 1 +s 0 1. The difference ~ f Λ 0 (h) f Λ 0 (h) Φ dt2 Φ ds 2 on U p D 2 " 1 is of the order of " 1, where D 2 " 1 denotes the " 1 -ball in R 2 centered at the origin. Adding f 2 1 dψ2 1 (twisting): Consider the C1 -map ff : R! R 2 dened by u 7! (" 1 cos " 1 u; " 1 1 sin " 1 u). Note that the map ff is an isometry with respect to the canonical 1 4

5 Euclidean metrics on R and R 2. Moreover, it takes R into the " 1 -ball D 2 " in R 2 centered at the origin. Dene a map ff 1 : U p! U p R 2 as follows: u 7! (u; " 1 f 1 (u) cos " 1 1 ψ 1(u); " 1 f 1 (u) sin " 1 1 ψ 1(u)). It can be shown that ff Λ 1 (dt2 Φ ds 2 )=f 2 1 dψ2 + " 2 1 df2. Since f 1 has its support contained in U p, the composite map ff μf 1 : U 1 p! Up R 2 f ~ 0! R q extends to a C 1 -map on all of V such that it coincides with f 0 outside U 1. We shall denote this extension also by μ f 1. Then μf Λ 1 (h) =f Λ 0 (h)+f2 1 dψ2 1 + O(" 1): The difference μ f Λ 1 (h) f Λ 0 (h) f2 1 dψ2 1 can be made arbitrarily small in the C 0 topology by controlling " 1. On the other hand it is clear from the construction that the C 0 -distance between f μ 1 and μf 0 = f 0 is less than " 1. Moreover, if fy 1 ;y 2 ;:::;y n g is a coordinate system on U p then for all j =1; f μ f μ 1 = f 1 + O(" 1 ) Repeating the above two steps for f 2 2 dψ2 2 on μ f 1 and so on in succession, we obtain a sequence of maps μ f 1 ; μ f 2 ;:::. Since U is locally nite and supports of only nitely many f i 's intersect an U p, the sequence f μ f i g is eventually constant on each U p. Therefore, lim μ f i exists uniformly on it. Let f 1 = lim μ f i. Then, f 1 is a C 1 map and f Λ 1 (h) =f Λ 0 (h)+ f 2 P i O(" i) which gives g f Λ 1 (h) = 1 2 (g f Λ 0 (h)) P i O(" i), where P i O(" i) can be made arbitrarily small in the ne C 0 topology. Also, on any U p 2U with coordinate functions fy 1 ;:::;y n g we f μ 0 = X f μ f μ i 1 = X i f i + X i O(" i ): Since at most N many f i 's (N depending on n) are non-vanishing at a point, we can apply the Cauchy-Schwartz inequality on the rst sum on the right hand side expression to obtain Therefore, ( P i jf i j) 2» N P i f2 i i ) 2» N P i f2 i dψ2 i )» N P i f2 i dψ2 i ) = N 2 ) ; kf 1 f 0 k 1» c(n)kfk P i O(" i). on each U p, where c(n) is a constant which depends on n. 5

6 We can choose " i at the i-th step for i =1; 2;::: in such away that f 1 is strictly short, and that the term P i O(" i) is strictly less than 1 6 kg f Λ 0 (h)k 0. This gives relation (ii) and (iii). We shall modify Nash's techniques to give a proof of Theorem 1.1. The initial map f 0 : V! R q in our context should be such that: (a) g 1 f Λ 0 h 1 and g 2 f Λ 0 h 2 are positive denite, and they have simultaneous decomposition as g 1 f Λ 0 h 1 = P i f2 i dψ2 i and g 2 f Λ 0 h 2 = P i c2 i f2 i dψ2 i for some C1 -functions f i and ψ i as above and some constants c i. (b) f 0 admits a `normal extension' f ~ 0 ; that is, f ~ Λ(h 0 1)=f Λ(h 0 1) Φ dt 2 Φ ds 2 f ~ Λ (h 0 2)=f Λ(h 0 2) Φ c 2 dt 2 Φ c 2 ds 2 on V for some specic constant c which appear in the decomposition. 3. Decomposition Lemma In this section and in the two subsequent ones we shall make the preparation for the Nash process. Here we state and prove the decomposition lemma in our context. We rst x a countable covering U = fu p jp 2 Zg of the manifold V which has the following properties: (a) each U p is a coordinate neighbourhood in V and (b) for any p 0, U p0 intersects atmost m different U p 's including itself (m is an integer depending on n = dim V ). Let ff p g be a smooth partition of unity on V subordinate to this covering fu p g such that support of f p, denoted supp f p, is compact for each p. Let K p = supp f p ρ U p. Lemma 3.1. Let g 1 and g 2 be two Riemannian metrics on V such that a 2 g 1 <g 2 <b 2 g 1, where a; b are two numbers with 0 <a<b. Then there exists a decomposition of g 1 and g 2 as: g 1 = 1X i=1 f 2 i (dψ i ) 2 ; g 2 = 1X i=1 c 2 i f2 i (dψ i ) 2 ; where for each i, f i and ψ i are C 1 functions on V with supp f i contained in some U p and c i 's are real numbers such that a<c i <b. Moreover, almost all f i 's vanish on an U p and there are at most N many f i which are simultaneously non-zero at any point of V, where N is an integer depending on n. Proof. Let P denote the vector space of quadratic forms on R n. Then dim P is n(n +1)=2. Let μ P be the subset of P P consisting of ordered pairs of positive denite quadratic forms (ff; ) such that a 2 ff<<b 2 ff. Then μ P is an open convex subset of P P. We cover μ P by geometrical open simplexes which lie completely within it. The collection of these gemetrical simplexes is locally nite and is such that no point lies in more than N 1 simplexes, where N 1 is a number depending on n. Now, since a 2 g 1 < g 2 < b 2 g 1, (g 1 ;g 2 ) maps K p onto a compact set in μ P and hence the image intersects only a nite number of simplexes, say ff 1 ;ff 2 ;:::;ff np. Denote the vertices of 6

7 ff μ, μ =1; 2;:::;n p, by (ff μ1 ; μ1 );:::;(ff μn2 ; μn2 ) where N 2 = n(n +1)+1. If (g 1 (x);g 2 (x)) lies in the interior to some simplex ff i then we can express (g 1 ;g 2 ) at x as (g 1 (x);g 2 (x)) = P ν c μν(x)(ff μν ; μν ), where c μν (x) > 0 for each ν and P ν c μν(x) =1. We introduce, for each μ =1; 2;:::n p,aweight function! μ by the following formula:! μ = expf P k 1=c μkg Pj expf P k 1=c jkg : Observe that! μ is a smooth function on a neighbourhood of K p and P μ! μ =1. These functions distribute a point over all the simplices in which it lies in the interior. Let c Λ μν = c μν! μ, μ = 1;:::n p and ν = 1;:::;N 2. Then it may be easily seen that g 1 = P μ;ν cλ μνff μν and g 2 = P μ;ν cλ μν μν on a neighbourhood of K p. Now for each pair (ff μν ; μν ) we can obtain a set of n-vectors V 1 μν;:::;v n μν with respect to which ff μν and μν can be simultaneously diagonalised in such away that ff μν = P n r=1 (dψr μν) 2 and μν = P n r=1 (vr μν) 2 (dψ r μν) 2, where v 1 μν ;:::;vn μν are n numbers, and ψ r μν is the linear functional dened on (U p ; y 1 ;y 2 ;:::;y n ) by ψ r μν = V r μν:(y 1 ;:::;y n ). Clearly a<v r μν <bfor all r =1; 2;:::;n. Now (g 1 ;g 2 )=( P p f pg 1 ; P p f pg 2 ). Since supp f p = K p the following expressions make sense all over V : Similarly, f p g 2 = f p Pμ;ν cλ μν μν = f p Pμ;ν cλ μν P n r=1 (vr μν) 2 (dψ r μν) 2 = P μ;ν f pc Λ μν P n r=1 (vr μν) 2 (dψ r μν) 2 = P n r=1 Pμ;ν (vr μν) 2 (f r μν) 2 (dψ r μν) 2 P n f p g 1 = r=1 Pμ;ν (fr μν) 2 (dψμν) r 2 where f r μν =(f p :c Λ μν) 1=2 are smooth functions on V which are supported in U p. Also, note that the functions ψ r μν in the above can be replaced by smooth functions which are dened all over V. Reindexing and then summing over p we get: g 1 = 1X p=1 X nnpn 2 r=1 (f r p) 2 (dψ r p) 2 g 2 = 1X p=1 nnpn X 2 r=1 (v r p) 2 (f r p) 2 (dψ r p) 2 where supports of the functions f r p are compact and contained in U p and a < v r p < b for all r =1; 2;:::;nn p N 2, p =1; 2;:::: Further this decomposition is such that there can be at the most mnn 1 N 2 terms which are non-vanishing at a given point of V. required features. This proves that the above decomposition has all the Remark 3.2. We assume g 1, g 2 to be C 1 for convenience only. If they are C 0 we can approximate them inside each U p by C 1 metrics which are sufciently C 0 close to g 1, g 2. It can be shown following Nash that the accumulated error at the end of each stage arising from this preliminary approximation can be controlled according to our requirement. 7

8 4. Existence of (g 1 ;g 2 )-short maps Denition 4.1. Let V be a manifold with two Riamannian metrics g 1 and g 2. A C 1 -map f 0 : V! (R q ;h 1 ;h 2 ) is (g 1 ;g 2 )-short if the metrics g 1 f Λ 0 (h 1) and g 2 f Λ 0 (h 2) on V are positive denite. This will be expressed by g i f Λ 0 (h i) > 0org i >f Λ 0 (h i), i =1; 2: Proposition 4.2. Let V be a C 1 -manifold with two Riemannian metrics g 1 and g 2 which are related by a 2 g 1 < g 2 < b 2 g 1 for some constants 0 < a < b. Then there exists a (g 1 ;g 2 )-short C 1 -immersion f 0 : V! (R q ;h 1 ;h 2 ) which also satises the following inequalities: (1) a 2 (g 1 f Λ 0 h 1 ) < (g 2 f Λ 0 h 2 ) <b 2 (g 1 f Λ 0 h 1 ): Proof. Here we prove the existence of f 0 when V is a compact manifold. This is done by scaling a given C 1 -immersion with a suitable scalar. Let f : be any immersion from V to R q. Consider the following four maps: F 1 ;F 2 : TV R! R dened respectively by G 1 ;G 2 : TV R! R dened respectively by for v 2 TV x, x 2 V, t 2 R. (v; t) 7! (g 1 t 2 f Λ h 1 ) and (v; t) 7! (g 2 t 2 f Λ h 2 ) (v; t) 7! (g 2 a 2 g 1 )(v; v) t 2 (f Λ h 2 a 2 f Λ h 1 )(v; v) and (v; t) 7! (b 2 g 1 g 2 )(v; v) t 2 (b 2 f Λ h 1 f Λ h 2 )(v; v) Let SV denote the sphere bundle associated to TV. Since by our hypothesis g 1 ;g 2 ;g 2 a 2 g 1 and b 2 g 1 g 2 are positive-denite, all the above maps, when restricted to SV f0g are strictly positive. Therefore, there exists a real-valued continuous function : V! R + such that F i (v; t) > 0, G i (v; t) > 0 for all v 2 TV x and t 2 (0; (x)), x 2 V, i =1; 2. If V is compact, the function has a global minima. Let it be called 0 which is strictly positive. Then the function f 0 = 0 :f : has all the desired property. The open case can be treated as in Nash[4] combining the above observation with a partition of unity argument. Remark 4.3. The set of (g 1 ;g 2 )-short immersions which also satisfy the inequality (1) is an open set in the ne C 1 -topology. 5. Regular Maps In this section we identify a class of maps which admit `normal extension' in our context (Section 6). We start with the following denition. 8

9 Denition 5.1. An immersion f : V! R q is said to be (h 1 ;h 2 ) regular or simply regular (when the Euclidean metrics h 1 and h 2 on R q are well understood from the context), if the h 1 - orthogonal complement ofdf x (TV x ) is transversal to the h 2 -orthogonal complement ofdf x (TV x ) at each x 2 V. Let A denote the linear transformation R q! R q dened by the relation: h 2 (w; w 0 ) = h 1 (Aw; w 0 ) for all w; w 0 2 R q. Then it can be shown easily that a map f : V! R q is regular if and only if df x (TV x ) Φ A(df x (TV x )) has the maximum dimension at every x 2 V. Proposition 5.2. A generic map f : V! R q is regular if q>3 dim V 1. Proof: Let dim V = n. Let ± be a subset of the Grassmannian manifold Gr n (R q ) which consists of all n-planes T in R q such that T A(T ) 6= 0. Then ± is the disjoint union of two sets ± 0 and ± 00 where 1. ± 0 consists of all n-planes T in R q which contains an eigenvector of A, 2. ± 00 consists of all n-planes T in R q which contains a 2-dimensional subspace spanned by a pair fv; Avg. The subset ± 0 of Gr n (R q ) is of dimension (n 1)(q n) since i 's are all distinct. On the other hand ± 00 is of dimension (q 1) + (n 2)(q n). Therefore, dim ± = max(dim ± 0 ; dim ± 00 ) = (q 1) + (n 2)(q n). Let R denote the open subset of J 1 (V;R q ) consisting of 1-jets of germs of immersions from V to R q. There is a canonical projection map p : R! Gr n (R q ) which maps an 1-jet j 1 f (x), x 2 V, onto the n-subspace Im df x in R q. Let f : V! R q be a C 1 -immersion and let jf 1 : V! J 1 (V;R q ) denote the 1-jet map of f. Then f is regular if p f jf 1 misses the set ±. If q>3n 1 then codim ± >n, and hence by the Thom Transversality Theorem p f jf 1 misses ± for a generic f. Thus a generic map f : V! R q is regular if q>3 dim V 1. Notation 5.3. For brevity we shall use the symbol T x for the subspace df x (TV x )inr q. T will denote the subbundle of f Λ (TR q ) whose bre over x is T x. The notation for the orthogonal complement of the subspace T x relative to the metric h i will be T? i x, i =1; 2. Denition 5.4. Let f : V! R q be a regular map. S Then the dimension of T? 1 x T? 2 x is q 2n for every x 2 V, where T x = df x (TV x ). Therefore, x2v T? 1 x T? 2 x denes a vector subbundle of f Λ (TR q ) of rank q 2n. of f. In the following, we denote this subbundle by BN and we call it the (h 1 ;h 2 )-normal bundle Remark 5.5. It follows from Remark 4.3 and Proposition 5.2 that any (g 1 ;g 2 )-short immersion f : V! R q satisfying the inequality a 2 (g 1 f Λ 0 h 1) < (g 2 f Λ 0 h 2) < b 2 (g 1 f Λ 0 h 1) can be approximated in the ne C 1 -topology by a regular (g 1 ;g 2 )-short immersion which satises the same inequality. 9

10 6. Normal Extension Let f : V! (R q ;h 1 ;h 2 ) be a C 1 -immersion which isregular on an open set U of V, and let a; b be two numbers satisfying the hypothesis of Theorem 1.1. We shall show that f admits a normal extension f ~ : U R 2! W over U such that ~f Λ (h 1 )=f Λ (h 1 ) Φ dt 2 Φ ds 2 and f ~ Λ (h 2 )=f Λ (h 2 ) Φ c 2 dt 2 Φ c 2 ds 2 on U f0g, for any constant c with a<c<b. Observe that to obtain such an extension we require two smooth vector elds and 0 along f in R q with the following properties: P1. and 0 are h 1 -aswell as h 2 -orthogonal to T (see Notation 5.3); P2. kk 1 = k 0 k 1 = 1 and kk 2 = k 0 k 2 = c; P3. and 0 are mutually orthogonal relative tobothh 1 and h 2. Indeed if such vector elds exist then we can dene f ~ : U R 2! R q by the formula f(x; ~ t; s) = f(x)+s + t 0. Now, P1 amounts to saying that and 0 are sections of the vector bundle BN over U. If U is contractible such elds always exist. Let h μ denote the pseudo-riemannian metric c 2 h 1 h 2 on R q, for a<c<b. The hypothesis of Theorem 1.1 implies that 1. h μ is non-singular; 2. the positive and the negative ranks of h μ are greater than or equal to 3n +2. Let h denote the quadratic form on BN induced from h, μ and let h x be the restriction of h to the bre BN x. If we denote the positive and the negative ranks of h x by q + (x) and q (x) respectively, then by our hypothesis q + (x) n + 2 and q (x) n + 2 for all x 2 V. Then the existence of and 0 satisfying P2 and P3 means P2 0. the vector elds and 0 are h-isotropic; that is, h(;) = 0 and h( 0 ; 0 )=0; P3 0. h(; 0 )=0. Hence and 0 dene in each bre of BN an h-isotropic 2-subspace. Before we proceed further, we set our terminology and recall some basic results. Denition 6.1. Let h be an indenite quadratic form on R q. Recall that the kernel or the null space of h consists of all those vectors v for which h(v; : ) 0. A subspace L in R q is called h-regular (or simply regular) if L ker h = 0 ([2], 2.4.9(B)). A subspace of R q which is regular as well as isotropic will be referred to as regular isotropic subspace. An isotropic vector which does not lie in the kernel of h will be called a regular isotropic vector. Given an indenite quadratic form h on R q, it may be recalled that the space of regular isotropic vectors has the same homotopy type as S q + 1 S q 1. Hence this space is k-connected for k = min(q + (h) 2;q (h) 2). In general, the space of regular isotropic m-subspaces 10

11 in R q has the homotopy type of V m (R q + ) V m (R q ) and is therefore (k 1)-connected for k» min(q + m; q m) ([2], 3.3.1(C)). Now we consider in the total space BN the subset which consists of all h x -regular isotropic vectors for all x 2 U and call it Iso. If Iso has a section, then normalizing this section relative to h 1 we get a section satisfying conditions P1 and P3. To obtain a pair of such sections which also satisfy P2 we consider the subset Iso 2 in BNΦBN dened as follows: Iso 2 = ρ (v; w) 2 BN Φ BN : span of v; w is a regular h isotropic 2-subspace ff : Lemma 6.2. Iso 2 is a submanifold in BN Φ BN and p Φ pj Iso2 is a submersion. Proof. Consider the 2-frame bundle of BN and denote it by V 2 (BN). An element (v; w) in V 2 (BN) which lies over x, will be denoted by (x; v; w). Further consider all those 2-frames (v; w) in the bres which span a h-regular subspace. We denote this set of vectors by V reg 2 (BN). Clearly this is an open subset in the total space BN Φ BN. Let f : V reg 2 (BN)! R 3 be the smooth map dened by f(x; v; w) =(h x (v; v);h x (w; w);h x (v; w)). Note that f 1 (0) is precisely the subset Iso 2. We shall show that f is a submersion. Let (x; v; w) 2 V reg 2 (BN). In order to prove that f is a submersion at this point itwould be enough to verify that the linear map f 0 : BN x Φ BN x! R 3 dened by (v 0 ;w 0 ) 7! df (x;v;w) (0;v 0 ;w 0 ) is surjective, since df (x;v;w) (u; v 0 ;w 0 )=df (x;v;w) (u; 0; 0) + df (x;v;w) (0;v 0 ;w 0 ). To compute df (x;v;w) (0;v 0 ;w 0 ) we need to calculate the derivative of the map (v; w) 7! (h x (v; v);h x (w; w);h x (v; w)). Observing that h x is a bilinear pairing we easily obtain: df (x;v;w) (0;v 0 ;w 0 )=(2h x (v; v 0 ); 2h x (w; w 0 );h x (v; w 0 )+h x (w; v 0 )). Now, the linear map (v 0 ;w 0 ) 7! (2h x (v; v 0 ); 2h x (w; w 0 )) has maximum rank if and only if v; w are regular vectors and ker h x (v; ) 6= ker h x (w; ). Note that if v and w are regular then, ker h x (v; ) = ker h x (w; ) if and only if there exists a 2 R such that v w is in ker h x, that is the span of v and w is not a regular subspace. Thus, if (x; v; w) 2 V reg 2 (BN) then (v 0 ;w 0 ) 7! (2h x (v; v 0 ); 2h x (w; w 0 )) is surjective. On the other hand since ker h x (v; ) 6= ker h x (w; ), there exists a v 0 2 BN x such that h x (v; v 0 ) = 0 and h x (w; v 0 ) 6= 0. Therefore, f 0 is surjective at each (x; v; w) 2 V reg 2 (BN). Hence, f is a submersion, and consequently f 1 (0) = Iso 2 is a submanifold of V reg 2 (BN). Finally, note that the tangent space to Iso 2 at a point (x; v; w) is given by the equation df (x;v;w) (u; v 0 ;w 0 ) = 0. Since df (x;v;w) restricted BN x Φ BN x is surjective, for any given u 2 TV x, there exists a (v 0 ;w 0 ) 2 BN x Φ BN x such that df (x;v;w) (0;v 0 ;w 0 )= df (x;v;w) (u; 0; 0). Clearly then (u; v 0 ;w 0 ) belongs to the tangent space of Iso 2. This proves that p Φ pj Iso2 11 is a

12 submersion. The bre of Iso 2 over x consists of all 2-frames which span a h x -regular isotropic subspace in the bre BN x. Hence it has the homotopy type of V 2 (R q + (x) ) V 2 (R q (x) ). By our hypothesis this is at least n 1 connected. Thus we obtain a section ( ; 0 ) of Iso 2 over U ρ V ([2], 3.3.1(A)). Then clearly and 0 satisfy the properties P1, P2 0 and P3 0. Moreover, span of and 0 denes a eld ff of two dimensional isotropic subspaces over U. We normalize the vector eld with respect to h 1 and call it. Since h 1 is positive denite,? 1 in R q intersects ff in a 1-dimensional subspace in each bre. This uniquely determines 0. Indeed, if lies in ff and has unit norm relative to h 1, then its h 2 norm is c. Also, if and 0 lie in ff and if they are orthogonal relative toh 1 then they are orthogonal relative toh 2 also. Thus we obtain vector elds and 0 over U which satisfy the properties P1, P2 and P3. 7. Proof of the Main Theorem We are now in a position to prove Theorem 1.1. We start with a (h 1 ;h 2 )-regular C 1 - immersion f 0 : V! R q which is(g 1 ;g 2 )-short and satises the inequality a 2 (g 1 f Λ h 0 1) <g 2 f Λ h 0 2 <b 2 (g 1 f Λ h 0 1) It follows from our discussion in Section 4 and 5 that such a map exists under the hypothesis a 2 g 1 <g 2 <b 2 g 1 and q 6n +4. We shall obtain a sequence of regular (g 1 ;g 2 )-short C 1 -immersions ff j g which satisfy the inequality (1) a 2 (g 1 f Λ j h 1 ) <g 2 f Λ j h 2 <b 2 (g 1 f Λ j h 1 ) and are such that (2) kg 1 f Λ j h 1k 0 < 2 3 kg 1 f Λ j 1 h 1k 0 ; kf j f j 1 k C 1 <c(n)kg 1 f Λ j 1 h 1k ; Then, the above three conditions would together imply that ff j g is a Cauchy sequence in the C 1 -topology and hence converges to a C 1 -map f : V! R q such that f Λ h 1 = g 1 and f Λ h 2 = g 2. Thus Theorem 1.1 will be proved. Observe that the sequence ff j g can be constructed inductively once we know how toget f 1 from the starting map f 0. Since f 0 is (g 1 ;g 2 )-short, both f 1 = g 1 f Λ 0 h 1 and f 2 = g 2 f Λ 0 h 2 are positive denite Riemannian metrics on V such that a 2 f 1 <f 2 <b 2 f 1. Therefore we can apply Lemma 3.1 for the pair (f 1 ;f 2 ) and obtain f 1 =2= 1X i=1 f 2 i dψ 2 i and f 2 =2= 1X i=1 c 2 i f 2 i dψ 2 i ; where f i and ψ i are smooth functions on V as described in the lemma, and c i 's are constants satisfying a<c i <bfor all i. 12

13 The map f 1 will be obtained through a sequence of smooth maps ff μ i g as lim i μ = f 1, where μf 0 = f 0, and μf Λ i (h 1)= f μλ (h i 1 1)+f 2 i dψ2 i + E i μf Λ i (h 2)= f μλ (h i 1 2)+c 2 i f2 i dψ2 i + : E0 i The error terms E i and E 0 i can be controlled simultaneously according to our requirement. The procedure of obtaining f μ i from f μ i 1 is the same for each i and so we only describe how to get μf 1 from f μ 0. Suppose supp f 1 ρ U p, where U p 2U, U being the covering of V described in Section 3. Since f 0 is regular and a < c 1 <b, where a; b satisfy the hypothesis of Theorem 1.1, we can get an extension ~ f 0 : U p R 2! R q of the map μ f 0 such that ~f Λ 0 (h 1)=f Λ 0 (h 1) Φ dt 2 Φ ds 2 and ~ f Λ 0 (h 2)=f Λ 0 (h 2) Φ c 2 1 dt2 Φ c 2 1 ds2 on U p f0g (see Section 6). Moreover, the difference ~ f Λ 0 (h) f Λ 0 (h) Φ dt2 Φ ds 2 on U p D 2 " is of the order of ", where D 2 " denotes the "-ball in R 2 centred at the origin. Now, proceeding exactly as in the proof of Lemma 2.2 we get a smooth map μ f 1 : V! R q such that μf Λ 1 (h 1)=f Λ 0 (h 1)+f 2 1 dψ2 1 + "2 1 df2 1 μf Λ 1 (h 2)=f Λ 0 (h 2)+c 2 1 f2 1 dψ2 1 + c2 1 "2 1 df2 1. The map μ f 1 obtained above may not be regular. However, a small perturbation near supp f 2 can make it regular there (as regularity is a generic property). Hence we can repeat the above steps on μ f 1 to get μ f 2 and so on in succession. The sequence f μ f i g then converges uniformly to a C 1 -map f 1 since only nitely many f i 's are non-vanishing on any open set U p. Now observe that g 1 f Λ 1 (h 1)= 1 2 (g 1 f Λ 0 (h 1)) P i O(" i) g 2 f Λ 1 (h 2)= 1 2 (g 2 f Λ 0 (h 2)) P i c2 i O(" i). We can choose " i at each stage in such away that (a) g 1 f Λ 1 (h 1) and g 2 f Λ 1 (h 2) are positive denite; (b) the error terms satisfy the inequalities: P i O(" i) < 1 6 kg 1 f Λ 0 (h 1)k 0 Then inequalities (2) follow from the discussion made in Section 2. Further, if we have P P i(b2 c 2 i )O(" i) < 1 2 [b2 (g 1 f Λ 0 h 1) (g 2 f Λ 0 h 2)] i (c2 i a2 )O(" i ) < 1 2 [(g 2 f Λ 0 h 2) a 2 (g 1 f Λ 0 h 1)] then f 1 also satises inequality (1). Now, by Remark 5.5 we can assume f 1 to be (h 1 ;h 2 )-regular. This completes the proof of Theorem 1.1 Acknowledgements. The authors would like to thank Professor Misha Gromov for his valuable comments on this paper. The second author is also thankful to the ICTP and IHES for their hospitality and other supports. 13

14 References [1] G. D'Ambra: Nash C 1 -embedding theorem for Carnot-Caratheodory metrics, Differential Geometry and its Applications 5(1995) [2] M. Gromov, Partial Differential Relations, Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge Band 9. Springer-Verlag, [3] N.H.Kuiper, C 1 -isometric embeddings, IProc. Koninkl. Nederl. Ak. Wet. A, 58(1955) [4] J.Nash, C 1 -isometric embeddings, Annals of Math. 60(1954)