Isometric Immersions with Flat Normal Bundle Between Space Forms

Home , Normal bundle

arXiv:2008.03929v1 [math.DG] 10 Aug 2020 etoa curvature sectional ie,i at h olwn lgtymr eea result: general more slightly following the fact, in gives, tn etoa curvature sectional connected stant simply complete a ihfltnra udeo opeeReana manifold Riemannian complete a of bundle normal flat with < c sphere al mesdit ulda pc ihfltnra ude Let bundle. normal flat b with cannot space curvature Euclidean into sectional immersed negative cally constant of manifolds nian h te ad epoe htteeeit naudneo loca of abundance an exists there that of proved immersions he hand, other the yHletfrtehproi ln.O n ad atn[]i 90sh 1920 in [4] Cartan hand, one On f result plane. would that hyperbolic classical conjecture the the the for of to Hilbert dimensions by answer higher positive to A generalization [8]. natural Gromov conjectured; and frequently been [10] has Moore immersion an such of existence eioerclyimre nteEcienspace Euclidean the in immersed isometrically be smti mesoswt a omlbundle normal flat with immersions Isometric fteeeit nioercimmersion isometric an exists there If ioaesy[1 rvdta opeennsml once Riema connected non-simply complete that proved [11] Nikolayevsky ti ogsadn rbe ftecmlt yeblcspace hyperbolic complete the if problem long-standing a is It ,rsetvl.I a bevdi 7 httepofb Nikolayevs by proof the that [7] in observed was It respectively. 0, H a a omlbnl,w rv htisscn udmna f fundamental second its that prove we exponentially. bundle, th grows If normal negative. flat is curvature curvatu has extrinsic negative the with that form so form space space a of immersion isometric an n S , c m eivsiaetebhvo ftescn udmna omo form fundamental second the of behavior the investigate We n rtehproi space hyperbolic the or ≥ .Djzr .R niadT.Vlachos Th. and Onti C.-R. Dajczer, M. ,cno eioerclyimre in immersed isometrically be cannot 3, H n into c with ewe pc forms space between R 2 n c < c − hti,teEcienspace Euclidean the is, that , 1 n htteehv l a omlbundle. normal flat all have these that and c ˜ m then , dmninlReana aiodo con- of manifold Riemannian -dimensional Abstract H c m 1 M codn owhether to according c n = f : H M c n . c n → R Q 2 R n n c ˜ − 2 + n 1 p − nfc,tenon- the fact, In . R , 2 m immersion e n vnlcly On locally. even h Euclidean the , M ≥ c eit a into re c 0 = n e a [12], Yau see 2 fconstant of Q isometri- e isometric l and orm o 1901 rom > c , c m H denote f < c n owed ea be or 0 can ky n- 0 , In view of Nikolayevsky’s result, the following weaker version of the problem discussed above has already been considered by Brander [3]. n PROBLEM: Do isometric immersions with flat normal bundle of Hc into n+p Qc˜ for n ≥ 2 and c< c˜ exist? In this paper, we analyze the behavior of the second fundamental form of a possible submanifold as in the problem above, and conclude that it must have exponential growth, as defined next. n n+p Let f : M → Qc˜ be an isometric immersion of a complete non-compact Riemannian manifold M n. It is said that the second fundamental form n αf : T M × T M → Nf M of f has exponential growth if there exist x0 ∈ M and positive constants k,ℓ ∈ R such that ℓr max {kαf (x)k : x ∈ Dr(x0)} ≥ ke for any r>r0 for some r0 > 0, where Dr(x0) denotes the closed geodesic n ball of M of radius r centered at x0 and kαf k is the norm of the second fundamental form given by 2 2 kαf (x)k = kαf (Xi,Xj)(x)k Xi,j where X1,...,Xn ∈ TxM is an orthonormal basis. n Theorem 1. If a complete n-dimensional Riemannian manifold Mc , n ≥ 2 n n+p and c < 0, admits an isometric immersion f : Mc → Qc˜ , c < c˜, with flat n n normal bundle then Mc = Hc and the second fundamental form of f has exponential growth. The above gives as corollary the result due to Bolotov [2] that there is n n+p no isometric immersion of Hc into R with mean curvature vector field of bounded length. The conclusion of Theorem 1 does not hold if the assumption of having flat normal bundle is dropped. For instance, it was shown by Aminov [1] that the example constructed by Rozendorn of an isometric immersion of H2 in R5 has no flat normal bundle and that the norm of its second fundamental form is globally bounded. The aforementioned result for codimension p = n − 1 due to Cartan has the following immediate consequence: n 2n−1 Corollary 2. If there exists an isometric immersion f : Hc → Qc˜ with c< c˜ then the second fundamental form of f has exponential growth.

2 1 The proof

n n+p n Let f : M → Qc˜ be an isometric immersion of a Riemannian manifold M n+p into the space form Qc˜ . If the immersion f has ﬂat normal bundle, that is, if at any point the curvature tensor of the normal connection vanishes, then it is a standard fact (cf. [7]) that at any point x ∈ M n there exists a set of unique pairwise distinct normal vectors ηi(x) ∈ Nf M(x), 1 ≤ i ≤ s(x), called the principal normals of f at x, and an associate orthogonal splitting of the tangent space as

TxM = Eη1 (x) ⊕···⊕ Eηs (x), where

Eηi (x)= X ∈ TxM : αf (X,Y )= hX,Y iηi for all Y ∈ TxM . n The multiplicity of a principal normal ηi ∈ Nf M(x) of f at x ∈ M is the dimension of the tangent subspace Eηi (x). If s(x) = k is constant on n n M , then the maps x ∈ M 7→ ηi(x), 1 ≤ i ≤ k, are smooth vector fields, called the principal normal vector fields of f. Moreover, also the distributions n x ∈ M 7→ Eηi (x), 1 ≤ i ≤ k, are smooth. n n+p In the sequel, let f : Mc → Qc˜ , c< c˜, be an isometric immersion with flat normal bundle. Since C =˜c − c> 0, it follows from the Gauss equation that any principal normal has multiplicity one. Thus, there exist exactly n nonzero principal normal vector fields η1,...,ηn satisfying

hηi, ηji = C, 1 ≤ i =6 j ≤ n. (1)

If Xi ∈ Γ(Eηi ), 1 ≤ i ≤ n, is a unit local vector ﬁeld then the local orthonormal frame X1,...,Xn diagonalizes the second fundamental form of f, that is, αf (Xi,Xj)= δijηi, 1 ≤ i, j ≤ n, where δij is the Kronecker delta. Such a frame is called a principal frame. Lemma 3. The following holds:

∇Xi Xj = −λiXj(1/λi)Xi, 1 ≤ i =6 j ≤ n, (2)

2 where λi =1 kηik + C. p 3 Proof: The Codazzi equation is equivalent to

⊥ ∇Xj ηi = h∇Xi Xi,Xji(ηi − ηj) (3) and

h∇Xℓ Xj,Xii(ηi − ηj)= h∇Xj Xℓ,Xii(ηi − ηℓ) (4) for all 1 ≤ i =6 j =6 ℓ =6 i ≤ n. The vectors ηi − ηj and ηi − ηℓ, 1 ≤ i =6 j =6 ℓ =6 i ≤ n are linearly independent. Suppose otherwise that ηi − ηj = µ(ηi − ηℓ). Taking the inner 2 product with ηi and using (1) gives kηik = C < 0, a contradiction. It now follows from (4) that

i j ∇Xi Xj =ΓijXi +ΓijXj, i =6 j,

k j where Γij = h∇Xi Xj,Xki. Since Γij = h∇Xi Xj,Xji = 0, then

i j ∇Xi Xj =ΓijXi = −ΓiiXi.

On the other hand, taking the inner product of (3) with ηi and using (1) is j easily seen to give that Γii = λiXj(1/λi), as we wished.

n Lemma 4. For each x0 ∈ Mc there exists a diﬀeomorphism F : U → V from n an open subset U ⊂ R endowed with coordinates {u1,...,un} onto an open n neigborhood V ⊂ Mc of x0 such that the tangent frame

2 2 kη1k + CF∗(∂/∂u1),..., kηnk + CF∗(∂/∂un) (5) p p n is orthonormal and principal. Moreover, if Mc is complete and simply con- n n nected then F : R → Mc is a diﬀeomorphism. Proof: For the local existence, observe that Lemma 3 implies

[λiXi,λjXj]=0, 1 ≤ i =6 j ≤ n.

For the proof of the global part, we follow a similar argument as in the n proof of Theorem 3 in [9] or Proposition 5.6 in [7]. Assume that Mc is n complete and simply connected. Set Yi = λiXi and let ϕi(x, t), x ∈ Mc , t ∈ R be the one-parameter group of diffeomorphisms generated by Yi. Since the vector fields Yi, 1 ≤ i ≤ n, have bounded lengths, it follows that ϕi(x, t) is n defined for all values of x and t. Thus, for any x ∈ Mc , the map t 7→ ϕi(x, t)

4 n is the integral curve of Yi with ϕi(x, 0) = x. Let x0 be a ﬁxed point in Mc n n and deﬁne a function F = Fx0 : R → Mc by

F (t1, t2,...,tn)= ϕn(ϕn−1(··· ϕ2(ϕ1(x0, t1), t2), ··· ), tn).

Since the Lie bracket [Yi,Yj] vanishes the parameter groups ϕi and ϕj com- mute. This implies that

F (t + s)= ϕ (ϕ − (··· ϕ (ϕ (F (s), t ), t ), ··· ), t )= F (t) (6) x0 n n 1 2 1 x0 1 2 n Fx0 (s) where t =(t1,...,tn) and s =(s1,...,sn). Thus d d F∗(s)∂ = | F (s ,...,s + t,...,s )= | ϕ (F (s), t)= Y (F (s)). i dt t=0 1 i n dt t=0 i i n We claim that F is a covering map. Then this and that Mc is simply connected yields that F is a diffeomorphism, which gives the proof. n ˜ Given x ∈ Mc , let B2ε(0) be an open ball of radius 2ε centered at the ˜ origin such that Fx|B˜2ε(0) is a diffeomorphism onto B2ε(x)= Fx(B2ε(0)). Set −1 {x˜α}α∈A = F (x) and denote by B˜2ε(˜xα) the open ball of radius 2ε centered atx ˜α. Define a map φα : B2ε(x) → B˜2ε(˜xα) by −1 φα(y)=˜xα + Fx (y). From (6) we obtain F (φ (y)) = F (˜x + F −1(y)) = F (F −1(y)) = F (F −1(y)) = y x0 α x0 α x Fx0 (˜xα) x x x ˜ for all y ∈ B2ε(x). Thus Fx0 is a diffeomorphism from B2ε(˜xα) onto B2ε(x) having φα as its inverse. In particular, this implies that B˜2ε(˜xα) and B˜2ε(˜xβ) are disjoint if α, β ∈ A are distinct indices. Finally, it remains to check that −1 ˜ ify ˜ ∈ Fx0 (Bε(x)), theny ˜ ∈ Bε(˜xα) for some α ∈ A. This follows from the fact that F (˜y − F −1(F (˜y))) = F (−F −1(F (˜y))) = x. x0 x x0 Fx0 (˜y) x x0 n For the last equality, observe from (6) that for all x, y ∈ Mc we have Fx(t)= y if and only if Fy(−t)= x. n The third fundamental form IIIf (x) of f at x ∈ M is given by

IIIf (X,Y )(x)=tr hαf (X, · ),αf (Y, · )i,X,Y ∈ TxM.

Since αf has no kernel (that is, positive index of relative nullity), then IIIf (x) is a positive deﬁnite inner product.

5 0 Lemma 5. The Riemannian metric g = Cg + IIIf is ﬂat where g is the n 0 metric of Mc . Moreover, the metric g is complete if g is complete.

Proof: In terms of the system of principal coordinates {u1,...,un} given by Lemma 4, we have

2 0 C kηik gij = Cgij + IIIf (∂/∂ui,∂/∂uj )= 2 δij + 2 δij = δij. kηik + C kηik + C

0 0 Moreover, the metric g is complete since gij >Cgij.

n n Proof of Theorem 1. By Nikolayevsky’s result we have that Mc = Hc . Let n n F : R → Hc be the global diﬀeomorphism given by Lemma 4. We endow Rn with the pullbacks of the two metrics considered in Lemma 5 that are still denoted by g and g0. Notice that (Rn, g0) is the standard ﬂat Euclidean space. Given a smooth curve γ :[a, b] ⊂ R → Rn set

2 Sˆ(γ) = max kαf k (F (γ(t))). t∈[a,b]

We have from (5) that 1 1 gij = 2 δij ≥ δij. kηik + C Sˆ(γ)+ C Then, the lengths of γ satisfy

1/2 Lg0 (γ) < (Sˆ(γ)+ C) Lg(γ). (7)

Let γ :[a, b] → Rn andγ ˜ :[a, b] → Rn be the unique Euclidean and hyperbolic geodesics, respectively, joining γ(a)=˜γ(a) to γ(b)=˜γ(b). From (7) we have ˆ 1/2 Lg0 (γ) ≤ Lg0 (˜γ) < (S(˜γ)+ C) Lg(˜γ). n Thus, if γx,y is the unique hyperbolic geodesic joining x =6 y ∈ R , then the distances with respect to g0 and g satisfy

ˆ 1/2 dg0 (x, y) < (S(γx,y)+ C) dg(x, y). (8)

n g g0 Fix x0 ∈ R and let Dr (x0) and Dr (x0) be the closed geodesic balls of 0 radius r> 0 centered at x0 with respect to g and g , respectively.

6 It holds that g g0 Dr (x0) ⊂ int D (x0) , (9) ψ(r) where ψ(r)= r(S(r)+ C)1/2 and S(r) = max (kα k2(F (x))). g f x∈Dr (x0)

g In fact, if y ∈ Dr (x0) we have using (8) that 1/2 1/2 0 ˆ ˆ dg (x0,y) < (S(γx0,y)+ C) dg(x0,y) ≤ r(S(γx0,y)+ C) ≤ ψ(r). Then, we obtain using (9) that the volumes of the geodesic balls satisfy

g g0 Volg(Dr (x0)) ≤ Volg Dψ(r)(x0) n 2 −1/2 = Πi=1(kηik + C) du1 ∧···∧ dun Z g0 Dψ(r)(x0)

−n/2 −n/2 g0 < C du ∧···∧ du = C Vol 0 D (x ) 0 1 n g ψ(r) 0 ZDg (x ) ψ(r) 0 n n/2 = r (1 + S(r)/C) ωn,

g where ωn is the volume of the Euclidean unit n-ball. Since Volg(Dr (x0)) is well known to grow exponentially with r (for instance, see [5]), it follows that also S(r) grows exponentially with r, and thus the second fundamental form of f has exponential growth. Remark 6. It is worth mentioning that it was shown in [6] that there is no isometric immersion with ﬂat normal bundle of a complete Riemannian n n+p manifold Mc , c > 0, into Qc˜ with c < c˜. Notice that this follows using Lemma 4.

References

[1] Aminov, Y., Extrinsic geometric properties of the Rozendorn surface, which is an isometric immersion of the Lobachevski plane into E5. Sb. Math. 200 (2009), 1575–1586. [2] Bolotov, D., On an isometric immersion with a ﬂat normal connection of the Lobachevsky space Ln into the Euclidean space Rn+m, Math. Notes 82 (2007), 10–12.

7 [3] Brander, D., Results related to generalizations of Hilbert’s non- immersibility theorem for the hyperbolic plane, Electron. Res. Announc. Math. Sci. 15 (2008), 8–16.

[4] Cartan, E., Sur les vari´et´es de courbure constante d’un espace euclidien ou non-euclidien, Bull. Soc. Math. France 47 (1919), 125–160; 48 (1920), 132–208.

[5] Chavel, I., “Riemannian Geometry: A Modern Introduction” Cambridge University Press, Cambridge, 1993.

[6] Dajczer M. and Tojeiro, R., Isometric immersions and the generalized Laplace and elliptic sinh-Gordon equations, J. Reine Angew. Math. 467 (1995), 109–147.

[7] Dajczer M. and Tojeiro, R., “Submanifold theory beyond an introduction”. Universitext. Springer, 2019.

[8] Gromov, M., Geometric, algebraic, and analytic descendants of Nash isometric embedding theorems, Bull. Amer. Math. Soc. 54 (2017), 173– 245.

[9] Moore, J. D., Isometric immersions of space forms in space forms, Paciﬁc J. Math. 40 (1972), 157–166.

[10] Moore, J. D., Problems in the geometry of submanifolds, Mat. Fiz. Anal. Geom. 9 (2002),648–662,

[11] Y. Nikolayevsky, Non-immersion theorem for a class of hyperbolic manifolds, Diﬀerential Geom. Appl. 9 (1998) 239–242.

[12] Yau, S.-T., Seminar on Diﬀerential Geometry, Ann. of Math. Stud. 102, Princeton Univ. Press, Princeton, N.J. (1982), 669–706,

Marcos Dajczer IMPA – Estrada Dona Castorina, 110 22460–320, Rio de Janeiro – Brazil e-mail: [email protected]

Christos-Raent Onti Department of Mathematics and Statistics

8 University of Cyprus 1678, Nicosia – Cyprus

Theodoros Vlachos University of Ioannina Department of Mathematics Ioannina – Greece e-mail: [email protected]