Existence of isometric immersions into nilpotent Lie groups Jorge H. S. de Lira∗ and Marcos F. Melo

†

Abstract We establish necessary and sufficient conditions for existence of isometric immersions of a simply connected Riemannian manifold into a two-step nilpotent Lie group. This comprises the case of immersions into H-type groups.

MSC 2000: 53C42, 53C30

1

Introduction

The fundamental theorem of submanifold theory, usually referred to as Bonnet’s theorem, states that the Gauss, Codazzi and Ricci equations constitute a set of integrability conditions for isometric immersions of a simply connected Riemannian manifold in Euclidean space with prescribed second fundamental form. From the viewpoint of exterior differential systems, this result is a classical application of Frobenius’s theorem. At this respect, we refer the reader to [5], [10] and [18] for instance. Versions of Bonnet’s Theorem for immersions in Riemannian spaces were recently achieved by Benoit Daniel in [6] and [7] and by P. Piccione and V. Tausk in [16] and [17]. In [7], Daniel consider immersions in three dimensional homogeneous spaces with four dimensional isometry group as Heisenberg spaces and Berger spheres. These ambients are regarded there as total spaces of Riemannian submersions over constant curvature surfaces, fibered by flow lines of a vertical Killing vector field ξ. It is proved that the ambient curvature tensor expressed in terms of a frame adapted to the immersion may be completely determined by the first and second fundamental forms and by the normal component ν and tangencial projection T of ξ. Since Gauss and Codazzi equations involve these projections, it is necessary to consider two additional first order differential equations in ν and T . The augmented set of equations is then a complete integrability condition. In [17], Piccione and Tausk prove a general existence result for affine immersions into affine manifolds endowed with a G-structure. The immersions should preserve the Gstructure and the ambient spaces are required to be infinitesimally homogeneous. Roughly ∗ †

partially supported by CNPq and FUNCAP partially supported by CAPES and CNPq

1

speaking, this last condition assures that the ambient curvature is constant when computed in terms of frames belonging to a G-reduction of the frame bundle. This method encompasses all classical results as well as Daniel’s results. Another applications of this technique in the context of Lie groups and Lorentzian Geometry may be found in [11], [12], [13] and [15]. The Heisenberg spaces studied in [7] are nilpotent Lie groups. Indeed, two-step nilpotent Lie groups form a distinguished class of geometric objects which include real, complex and quaternionic Heisenberg spaces and more generally H-type groups (see, e.g., [9], [8], [1] and [2]). These groups have some remarkable analytical properties and appear in distinct areas as Harmonic Analysis (v. [4]) and General Relativity (v. [14]). These remarks motivate us to raise the question of extending Bonnet’s theorem from the classical case, which corresponds to Abelian groups, to two-step nilpotent Lie groups. Theorem 1 below yields such an extension in the spirit of the results in [7]. A brief outline of this paper may be given as follows. Let N be a (n + n0 )-dimensional two-step nilpotent Lie group, where n0 is the dimension of the center z in its Lie algebra. As occurs in [7], z is spanned by left-invariant n0 Killing vector fields En+k , k = 1, . . . , n0 , whose covariant derivatives determine certain skew-symmetric tensors Jk , k = 1, . . . , n0 . The Section 2 is devoted to show that the curvature tensor in N may be computed in 0 k an arbitrary frame {ea }n+n a=1 solely in terms of the tensors Jk and the projections Ua = 0 hEn+k , ea i. The curvature form relative to the frame {ea }n+n a=1 is given by the tensor Q defined in Section 2.1.2. In the particular case of a frame adapted to an isometric immersion, this implies that Gauss, Codazzi and Ricci equations are completely written only in terms of the first and second fundamental forms and the normal and tangential projections of the Killing vector fields En+1 , . . . , En+n0 and their covariant derivatives. This is the content of Section 3. In Sections 4 and 5, we establish sufficient conditions for immersing isometrically a simply connected Riemannian manifold M into N , with prescribed second fundamental form. For this, we consider a real Riemannian vector bundle E over M with rank m0 = n + n0 − m so that the Whitney sum S = T M ⊕ E is a trivial bundle. We define an ˆa }n+n0 in S and then transplante the definitions of the tensors orthonormal global frame {E a=1 Jk and Q to this setting. This may be done in last analysis because these tensors depend on the structural constants of N . We then prove Theorem 1 a) Let M m be a Riemannian simply connected manifold and let E be a real Riemannian vector bundle with rank m0 so that S = T M ⊕ E is a trivial vector bundle. ˆ and R ˆ be respectively the compatible connection and curvature tensor in S and ∇ Let ∇ E and ∇ the compatible connections induced in T M and E, respectively. We fix a global ˆk }n+n0 in S. Define Jˆk and Q ˆ as in (4.1) and (4.4), respectively. orthonormal frame {E k=1 Assume that these fields satisfy the Gauss, Codazzi and Ricci equations ˆ=Q ˆ R 2

(1.1)

and the additional equations ˆE ˆn+k = −1/2 Jˆk , ∇

k = 1, . . . , n0 .

(1.2)

Thus, there exists an isometric immersion f : M → N covered by a bundle isomorphism f∗⊥ : E → T Mf⊥ , where T Mf⊥ is the normal bundle along f so that f∗⊥ is an isometry when restrited to the fibers and satisfies ⊥ f∗⊥ ∇EX V = ∇⊥ X ∈ Γ(T M ), V ∈ Γ(E), X f∗ V, ⊥ ¯ f X f∗ Y − f∗ (∇X Y ), X, Y ∈ Γ(T M ), f∗ II(X, Y ) = ∇ ∗

(1.3) (1.4)

¯ and ∇⊥ denote, respectively, the connections in N and T M ⊥ and the tensor where ∇ f II ∈ Γ(T ∗ M ⊗ T ∗ M ⊗ E) is defined by ˆ X Y = ∇X Y + II(X, Y ), ∇

X, Y ∈ Γ(T M ).

(1.5)

b) Let f, f˜ be two isometric immersions from M to N with second fundamental forms IIf and IIf˜ satisfying IIf (X, Y ) = ΦIIf˜(X, Y ), X, Y ∈ Γ(T M ), (1.6) ˜ ⊥ on the respective normal bundles T M ⊥ and T M ⊥ and normal connections ∇⊥ and ∇ f f˜ related by ˜⊥ (1.7) Φ∇⊥ V ∈ Γ(T Mf⊥ ), X V = ∇X Φ(V ), where Φ : T Mf⊥ → T Mf⊥ ˜ is a vector bundle isomorphism satisfying hΦ(V ), Φ(W )i = hV, W i,

V, W ∈ Γ(T Mf⊥ ).

(1.8)

0

Fixed a left-invariant frame {Ek }n+n k=1 in N we assume that hf∗ X, En+k i = hf˜∗ X, En+k i,

X ∈ Γ(T M )

(1.9)

V ∈ Γ(T Mf⊥ ).

(1.10)

and that hV, En+k i = hΦ(V ), En+k i, 1, . . . , n0 .

for k = Then, there exists an isometry L : N → N such that f˜ = L ◦ f . The ultimate reason for refer to (1.1) as Gauss, Codazzi and Ricci equations is that the ˆ imitates the curvature tensor in N when written in terms of a frame adapted to tensor Q an isometric immersion. We point out that imposing that S is trivial allows us to give an intrinsic meaning to the tensors Jˆk . Hypothesis (1.1) and (1.2) play here the same role as the construction of a flat bundle endowed with parallel sections in the proof of the classical case. Our method keeps some resemblance with the proof of Bonnet’s theorem given by P. Ciarlet and F. Larsonneur in [3]. Indeed, Theorem 1 may be regarded as establishing sufficient conditions for immersing an open set of the Euclidean space into a two-step nilpotent Lie group. 3

2

Two-step nilpotent Lie groups

Let N be a Lie group with Lie algebra n and Maurer-Cartan form ωn . The Levi-Civit` a connection of a given left-invariant metric h·, ·i on N is ¯ E F = [E, F ] − ad∗E · F − ad∗F · E, 2∇

(2.1)

where E, F are left-invariant vector fields in n and had∗E · F, Gi = hF, [E, G]i,

E, F, G ∈ n.

We suppose that n may be decomposed as n = z ⊕ v with [v, v] ⊂ z,

[z, n] = {0},

(2.2)

what implies that N is a two-step nilpotent Lie group. Let us denote by n and n0 the dimensions of v and z, respectively. We suppose that the direct sum n = z⊕v is orthogonal. The relations (2.2) then yield ¯ E F = 1 [E, F ], E, F ∈ v, ∇ 2 ¯ EZ = ∇ ¯ Z E = − 1 JZ E, E ∈ v, Z ∈ z, ∇ 2 ¯ Z Z 0 = 0, Z, Z 0 ∈ z, ∇

(2.3) (2.4) (2.5)

where the operator JZ : v → v associated to a vector field Z ∈ z is defined by JZ = ad∗ Z. This operator may be extended to the whole algebra n as ¯ JZ := −2∇Z.

(2.6)

It is useful to consider also the (0, 2) tensor field equally denoted by JZ and defined by JZ (E, F ) = hJZ E, F i.

2.1

Some auxiliary tensors

According to the decomposition n = v ⊕ z, we choose an orthonormal left-invariant frame field E1 , . . . , En , En+1 , . . . , En+n0 , (2.7) so that the first n vectors are in v and the next n0 ones are in z. Fixed this choice of frame, we define the structural constants of N by [Ek , El ] =

0 n+n X

r σkl Er ,

r=1

4

1 ≤ k, l ≤ n + n0 .

(2.8)

0

0

n+n If {θk }n+n k=1 denotes the co-frame dual to {Ek }k=1 , then the corresponding connection forms in N are given by n+n0 1 X k r k θl = τlr θ , (2.9) 2 r=1

where k l r τlrk = σrl + σkr + σkl .

We also define the curvature 2-form Θ =

n+n0 {Θkl }k,l=1

(2.10)

of N associated to (2.7) by

0

Θkl

n+n 1 X = 4


r k r s τlrk τst + τrs τlt θ ∧ θt .

(2.11)

r,s,t=1

These forms satisfy the structural equations k

dθ +

0 n+n X

θlk ∧ θl = 0,

θlk = −θkl

(2.12)

l=1

and dθlk

+

0 n+n X

θrk ∧ θlr = Θkl ,

(2.13)

r=1

where 1 ≤ k, l ≤ n + n0 . 2.1.1

Christoffel tensor

Given the left-invariant frame above, we denote Jk = JEn+k , 1 ≤ k ≤ n0 . Fixed this notation, we define in N the tensor field n0

n0

k=1

k=1

1X 1X L(X, Y, V ) = − hJk V, XihY, En+k i + hJk Y, XihV, En+k i 2 2 −

1 2

n0 X

hJk Y, V ihX, En+k i,

X, Y, V ∈ Γ(T N ).

(2.14)

k=1 0

In order to derive a local expression for L, we consider a frame {ea }n+n a=1 defined in an open 0 set N of N by 0 n+n X ea = Eb Aba , (2.15) b=1

for some map A : functions

N0

→ SOn+n0 . For 1 ≤ a ≤ n + n0 and 1 ≤ k ≤ n0 , we define the Uak = θn+k (ea ) = An+k . a 5

(2.16)

0

0

b n+n Thus, if (ω a )n+n a=1 and (ωa )a,b=1 are respectively the dual forms and the connection forms 0

associated to the frame {ea }n+n a=1 , one has ¯ n+k ) = dUak − ω a (∇E

X

Uck ωac =:

c

1X k b uab ω . 2

(2.17)

b

Hence, one gets 0 n+n X

Jk =

ukab ω a ⊗ ω b .

(2.18)

a,b=1

Notice that ¯ V En+k , W i = −2 hJk V, W i = −2h∇

X

¯ E En+k , Er i hV, El ihW, Er ih∇ l

l,r

X

=

n+k . hV, El ihW, Er iσlr

(2.19)

l,r

In local terms, that is, setting V = ea , W = eb , one has ukab =

n X

n+k . Ala Arb σlr

(2.20)

l,r=1

Using the local frame, one computes n0

n0

1X 1X L(X, ea , eb ) = − hJk eb , Xihea , En+k i + hJk ea , Xiheb , En+k i 2 2 k=1

k=1

0

−

n 1X

2

hJk ea , eb ihX, En+k i

k=1 0

0

n+n n
1 XX k k Ua ubc − Ubk ukac + Uck ukab ω c (X). = − 2 c=1 k=1

0

n+n One then defines the matrix of 1-forms λ = (λab )a,b=1 by

λab = L( · , ea , eb ),

(2.21)

that is, 0

λab

0

n+n n
1 XX k k =− Ua ubc − Ubk ukac + Uck ukba ω c . 2

(2.22)

c=1 k=1

We now use the equation (2.20) for obtaining an alternative expression for λ, which will be useful later. 6

0

Proposition 1 The 1-form λ = (λab )n+n a,b=1 defined in (2.21) satisfy λ = A−1 θA,

(2.23)

0

0

a n+n where θ = (θlk )n+n k,l=1 . Thus, the connection form ω = (ωb )a,b=1 is given by

ω = A−1 dA + λ.

(2.24)

Proof. Using (2.1) and (2.10), one obtains  k  σrl , 1 ≤ l, r ≤ n and k ≥ n + 1, σ r , 1 ≤ k, l ≤ n and r ≥ n + 1, τlrk =  kl l , 1 ≤ k, r ≤ n and l ≥ n + 1. σkr Thus, (2.22) and (2.20) yield 0

λab

0

n+n n 1 XX k k = − (Ua ubc − Ubk ukac − Uck ukab )ω c 2 c=1 k=1

0 n0 n+n XX

1 = − 2 +

n X

0

n+k c ω Uak Alb Arc σlr

c=1 k=1 l,r=1

1 2

0 n0 n+n XX

n X

0

n+n n n 1 X X X k l r n+l c + Aa Ub Ac σkr ω 2 c=1 l=1 k,r=1

n+r c ω Aka Alb Ucr σkl

c=1 r=1 k,l=1

what implies that λab

=

=

1 2

0 n+n X

0

Aka Alb Arc τlrk ω c

c,k,l,r=1

0 n+n X

k,l,r=1

k,l=1

(A−1 )ak θlk Alb

k,l=1 −1

= (A

0

n+n n+n X 1 X k l k r = Aka Alb θlk Aa Ab τlr θ = 2

θA)ab

This finishes the proof of the proposition. 2.1.2



A curvature-type tensor

We then define a (0, 4) covariant tensor Q in N by Q(X, Y, V, W ) = Q1 (X, Y, V, W ) + Q2 (X, Y, V, W ),

7

(2.25)

where X, Y, V and W are vector fields in N and Q1 and Q2 are given by Q1 (X, Y, V, W ) 1 1 1 = hJk X, W ihJk V, Y i + hJk Y, XihJk W, V i − hJk Y, W ihJk V, Xi 4 2 4 X 1X 1 ¯ X Jk )V, Y i + ¯ X Jk )W, Y i − hW, En+k ih(∇ hV, En+k ih(∇ 2 2 k k X 1 1X ¯ X Jk )W, V i + ¯ Y Jk )V, Xi hY, En+k ih(∇ hW, En+k ih(∇ + 2 2 k k X 1 1X ¯ Y Jk )W, Xi − ¯ Y Jk )W, V i hV, En+k ih(∇ hX, En+k ih(∇ − 2 2 k

k

and Q2 (X, Y, V, W ) 1X 1X =− hEn+k , W ihEn+l , V ihJk Y, Jl Xi + hEn+k , W ihEn+l , XihJk Y, Jl V i 4 4 k,l

k,l

1X 1X hEn+k , Y ihEn+l , V ihJk W, Jl Xi + hEn+k , Y ihEn+l , XihJk W, Jl V i − 4 4 k,l

+ +

1X 4

k,l

hEn+k , W ihEn+l , V ihJk X, Jl Y i −

k,l

1X 4

hEn+k , W ihEn+l , Y ihJk X, Jl V i

k,l

1X 1X hEn+k , XihEn+l , V ihJk W, Jl Y i − hEn+k , XihEn+l , Y ihJk W, Jl V i. 4 4 k,l

k,l

An important relation between λ and Q is given by the following lemma Lemma 1 The components Qba of Q are given by the 2-forms Qab := Q( · , · , eb , ea ) = dλ + λ ∧ ω + ω ∧ λ − λ ∧ λ


a

b.

(2.26)

Proof. Denoting the right hand side in (2.26) by Λab and expanding it, it results that X XX X X Ubk ωcb )ukad −2Λad = (dUak − Ubk ωab )ukdc − (dUdk − Ubk ωdb )ukac − (dUck − c

k

+ Uak (dukdc

b

−

X

−

X

−

X

ukdb ωcb

− −

X

−

X

b

− Udk (dukac

X b

ukbc λbd

−

ukbc ωab )

b

ukab ωdb

b

+ Uak

b

ukbc ωdb )

b

ukab ωcb

b

− Uck (dukad

b

X

ukbd ωab )

b

X b

Ubk λbd ukac

− Uck

X b

8


ukab λbd ∧ ω c .

(2.27)

The covariant derivative of the (0, 2) tensor Jk has components given in terms of the frame 0 {ea }n+n a=1 by ¯ k (ea , eb ) = duk − uk ωad − uk ω d =: ∇u ¯ k. ∇J ab db ad b ab

(2.28)

Using (2.17) and (2.28), one gets
0 1 1 1 − ukc0 a ukdc + ukc0 d ukac + ukc0 c ukad ω c ∧ ω c 2 2 2 k,c,c0 X
¯ k − U k ∇u ¯ k − U k ∇u ¯ k ∧ ωc Uak ∇u + ac c dc d ad X

−2Λad =

k,c

+

X

Uak

k,c

X

ukbc λbd −

b

X

Ubk λbd ukac − Uck

b

X


ukab λbd ∧ ω c .

b

The last three terms may be calculated using that for 1 ≤ k ≤ n0 , 1 ≤ a ≤ n + n0 , one has X X dUak − Uck ωac + Uck λca = 0. (2.29) c

c

For proving (2.29), using (2.14), one computes X X Uck λca = Uck L(·, ec , ea ) = L(·, En+k , ea ) c

c

1 X hJl ea , ·ihEn+k , En+l i − hJl En+k , ·ihea , En+l i 2 l  +hJl En+k , ea ih·, En+l i X X
1 hJl En+k , ·ihea , En+l i − hJl En+k , ea ih·, En+l i . = − hJk ea , ·i − 2

= −

l

l

However, given any vector field V in N , one has X X n+l n+l hJl En+k , V i = hEn+k , Er ihV, Es iσrs = hV, Es iσn+k,s = 0. r,s

s

Therefore, one concludes that X c

X 1 1X k b Uck λca = − hJk ea , ·i = − uab ω = dUak − Uck ωac , 2 2 c b

9

as desired. This proves (2.29). Thus, we may write
0 1 1 − ukc0 a ukdc + ukc0 c ukad ω c ∧ ω c 2 2 k,c,c0 X
¯ kac − Uck ∇u ¯ k ∧ ωc ¯ k − U k ∇u + Uak ∇u ad dc d

−2Λad =

X

k,c

+

X

Uak

X

ukbc λbd − Uck

b

k,c

X


ukab λbd ∧ ω c .

b

Nevertheless, in view of (2.14), it follows that X X X
Uak ukbc λbd + Uck ukba λbd = hEn+k , ea ihJk eb , ec i + hEn+k , ec ihJk eb , ea i L(·, eb , ed ) b

b

b

1X 1X hEn+k , ea ihEn+l , ed ihJk ec , Jl ·i − hEn+k , ea ihEn+l , ·ihJk ec , Jl ed i = 2 2 l l 1X 1X + hEn+k , ec ihEn+l , ed ihJk ea , Jl ·i − hEn+k , ec ihEn+l , ·ihJk ea , Jl ed i. 2 2 l

l

Therefore, one concludes that Λad =

X 1
0 1 ukc0 a ukdc − ukc0 c ukad ω c ∧ ω c 4 4 0

k,c,c

−


c0 1 X c k¯ k ¯ k ¯ c0 u k − U k ∇ Uak ∇ d c0 uac − Uc ∇c0 uad ω ∧ ω dc 2 0 k,c,c

−

X k,l,c,c0

1 1 hEn+k , ea ihEn+l , ed ihJk ec , Jl ec0 i − hEn+k , ea ihEn+l , ec0 ihJk ec , Jl ed i 4 4


0 1 1 + hEn+k , ec ihEn+l , ed ihJk ea , Jl ec0 i − hEn+k , ec ihEn+l , ec0 ihJk ea , Jl ed i ω c ∧ ω c , 4 4 what finishes the proof of the lemma.



This lemma has the following consequence, which characterizes geometrically the tensor Q. Proposition 2 The tensor Q satisfies the equation Q = A−1 ΘA 0

where Θ = (Θkl )n+n k,l=1 are the curvature forms defined in (2.11).

10

(2.30)

Proof. One has dω + ω ∧ ω = d(A−1 dA) + A−1 dA ∧ A−1 dA + dλ + λ ∧ λ + λ ∧ A−1 dA + A−1 dA ∧ λ = −A−1 dA ∧ A−1 dA + A−1 dA ∧ A−1 dA + d(A−1 θA) + A−1 θ ∧ θA +A−1 θ ∧ dA + A−1 dAA−1 ∧ θA = dA−1 ∧ θA + A−1 dθA − A−1 θ ∧ dA + A−1 θ ∧ θA +A−1 θ ∧ dA − dA−1 ∧ θA = A−1 (dθ + θ ∧ θ)A = A−1 ΘA.

(2.31)

On the other hand we have d(ω − λ) + (ω − λ) ∧ (ω − λ) = −A−1 dA ∧ A−1 dA + A−1 dA ∧ A−1 dA = 0,

(2.32)

what implies that dω + ω ∧ ω = dλ − λ ∧ λ + ω ∧ λ + λ ∧ ω. Hence (2.31) and (2.33) give the desired result.

3

(2.33) 

Isometric immersions into two-step nilpotent Lie groups

From now on, we consider a simply connected Riemannian manifold M m . We denote m0 = n + n0 − m. From the calculations above, we infer the following necessary conditions for the existence of isometric immersions in N with prescribed second fundamental form. In the statement, ¯ denotes the curvature tensor in N . R Proposition 3 Let f : M → N be an isometric immersion. Then, the Gauss, Ricci and Codazzi equations are given by ¯ ∗ X, f∗ Y, V, W ) = Q(f∗ X, f∗ Y, V, W ), R(f

X, Y ∈ Γ(T M ), V, W ∈ Γ(f ∗ T N ).

(3.1)

Moreover, the following additional equations are satisfied ¯ X En+k = − 1 Jk X, ∇ 2

X ∈ Γ(T M )

(3.2)

for k = 1, . . . , n0 . Proof. After identifying M and the immersed submanifold f (M ) ⊂ N , we consider an 0 orthonormal frame {ea }m+m defined in an ambient open neighborhood of an arbitrary a=1 point in M . This frame may be chosen adapted to the immersion, that is, in such a way 11

that, along points in M , the first m fields in this frame are tangent to M and the other m0 ones are local sections of the normal bundle T Mf⊥ . 0

Let A be given as above by (2.15). Then, the connection forms {ωba }m+m a,b=1 satisfy dωba +

X

ωca ∧ ωbc = (A−1 ΘA)ab ,

(3.3)

c

where Θ is given by (2.11). Since the right-hand side in (3.3) corresponds to the ambient curvature tensor expressed in terms of the adapted frame, this equation corresponds to Gauss, Codazzi and Ricci equations, respectively, as we may easily verify considering suitable ranges of indices a, b. Hence, (2.30) in Proposition 2 implies (3.1). The equation (3.2) follows immediately from the preceding discussion. 

4

Existence of an adapted frame

We now consider a real Riemannian vector bundle E over M with rank m0 and the Whitney ˆ and R ˆ be sum bundle S = T M ⊕ E. The metric in S is also represented by h·, ·i. Let ∇ respectively the compatible connection and curvature tensor in S. We suppose that S is a trivial vector bundle and then we fix a globally defined orthonorˆ1 , . . . , E ˆn+n0 in S. Hence, for any k = 1, . . . , n0 , one defines mal frame E hJˆk V, W i =

n X

ˆl ihW, E ˆr iσ n+k , hV, E lr

V, W ∈ Γ(S),

(4.1)

l,r=1 n+k where the constants σlr are given by (2.8). It is obvious from the definition that

ˆn+l i = 0 hJˆk V, E

(4.2)

n+k = 0. since σr,n+l

ˆ and Q ˆ in S by Now, we define in terms of Jˆk tensors L 0

1 ˆ L(X, Y, V ) = − 2 −

1 2

n X k=1

n0

X ˆn+k i + 1 ˆn+k i hJˆk V, XihY, E hJˆk Y, XihV, E 2

n0 X

k=1

ˆn+k i, hJˆk Y, V ihX, E

X, Y, V ∈ Γ(S)

(4.3)

k=1

and for X, Y ∈ Γ(T M ) and V, W ∈ Γ(S), ˆ ˆ 1 (X, Y, V, W ) + Q ˆ 2 (X, Y, V, W ), Q(X, Y, V, W ) = Q

12

(4.4)

where ˆ 1 (X, Y, V, W ) Q 1 1 1 = hJˆk X, W ihJˆk V, Y i + hJˆk Y, XihJˆk W, V i − hJˆk Y, W ihJˆk V, Xi 4 2 4 X 1X 1 ˆn+k ih(∇ ˆ X Jˆk )V, Y i + ˆn+k ih(∇ ˆ X Jˆk )W, Y i − hW, E hV, E 2 2 k k X 1 1X ˆn+k ih(∇ ˆ X Jˆk )W, V i + ˆn+k ih(∇ ˆ Y Jˆk )V, Xi hY, E hW, E + 2 2 k k X 1X 1 ˆn+k ih(∇ ˆ Y Jˆk )W, Xi − ˆn+k ih(∇ ˆ Y Jˆk )W, V i − hV, E hX, E 2 2 k

k

and ˆ 2 (X, Y, V, W ) Q X 1X ˆ ˆn+l , V ihJˆk Y, Jˆl Xi + 1 ˆn+k , W ihE ˆn+l , XihJˆk Y, Jˆl V i hEn+k , W ihE hE =− 4 4 k,l

−

1X 4

k,l

ˆn+k , Y ihE ˆn+l , V ihJˆk W, Jˆl Xi + hE

k,l

1X 4

ˆn+k , Y ihE ˆn+l , XihJˆk W, Jˆl V i hE

k,l

X 1X ˆ ˆn+l , V ihJˆk X, Jˆl Y i − 1 ˆn+k , W ihE ˆn+l , Y ihJˆk X, Jˆl V i + hEn+k , W ihE hE 4 4 k,l

+

1X 4

k,l

ˆn+k , XihE ˆn+l , V ihJˆk W, Jˆl Y i − hE

k,l

1X 4

ˆn+k , XihE ˆn+l , Y ihJˆk W, Jˆl V i. hE

k,l

We then suppose that ˆ ˆ hR(X, Y )V, W i = Q(X, Y, V, W ),

X, Y ∈ Γ(T M ), V, W ∈ Γ(S).

(4.5)

We also assume the following condition ˆ XE ˆn+k = − 1 Jˆk X, ∇ 2

X ∈ Γ(T M ),

k = 1, . . . , n0 .

(4.6)

The connection in S induces connections ∇ in M and ∇E in E. More precisely, defining II ∈ Γ(T ∗ M ⊗ T ∗ M ⊗ E) by ˆ X Y = ∇X Y + II(X, Y ), ∇

X, Y ∈ Γ(T M )

(4.7)

and defining, for V ∈ Γ(E), hSV (X), Y i = hII(X, Y ), V i, 13

(4.8)

one obtains

ˆ X V = −SV X + ∇EX V. ∇

(4.9)

ˆn+k = Tk + Nk , Tk ∈ Γ(T M ), Nk ∈ Γ(E), the condition In terms of the decomposition E (4.6) becomes 1 ∇X Tk − Sk (X) + ∇EX Nk + II(Tk , X) = − Jˆk (X), 2

X ∈ Γ(T M ),

(4.10)

where Sk = SNk . Definition 1 Given a connected simply connected open subset M 0 ⊂ M , we fix a map ˆ ∈ C ∞ (M 0 , Rn0 (n+n0 ) ). A frame e : M 0 → F(S) with components U e1 , . . . em , em+1 , . . . em+m0 is admissible if the first m sections are vector fields in M 0 and the last m0 ones are sections in E and, moreover, if it holds that 1 ≤ k ≤ n0 .

ˆn+k , ea i = U ˆak , hE

(4.11)

In particular, this implies that ˆn+k , ei i = U ˆik hTk , ei i = hE

(4.12)

for i = 1, . . . , m and ˆn+k , eα i = U ˆk hNk , eα i = hE α

(4.13) n+n0

ˆk } for α = m + 1, . . . , m + m0 . The transition map from the frame {E k=1 to an admissible 0 m+m frame {ea }a=1 is given by an admissible map, that is, if ea =

0 n+n X

ˆk Aka , E

(4.14)

k=1

then A is if the form  A(x) =

∗ ˆ U (x)

 ,

(4.15)

ˆ (x) corresponds to the last n0 lines. where the block U We denote by

0

ω 1 , . . . , ω m , ω m+1 , . . . , ω m+m

14

(4.16)

0 ˆ the real-valued 1-forms dual to the frame {ea }m+m a=1 . The Riemannian connection ∇ is given 0 n+n in terms of this frame by the matrix ω = (ωba )a,b=1 . Hence, the first structural equation is written as X dω a + ωba ∧ ω b = 0, ωba = −ωab . (4.17)

b

Regarding each Jˆk as (0, 2) tensor, we write them locally as X Jˆk = u ˆkab ω a ⊗ ω b , k = 1, . . . , n0 .

(4.18)

a,b

Thus in local terms the equation (4.6) is rewritten as X X X
ˆ k ωab = 1 ˆak − U dU u ˆkab ω b . b 2 k

b

(4.19)

k

ˆ is given by the 1-forms The local expression for L ˆ a = L( ˆ · , ea , eb ). λ b

(4.20)

Following the calculations in Section 2.1.1, we conclude that 0

ˆa λ b

0

n+n n
1 X X ˆk k k ˆ k ˆk ω c . ˆ ku Ua u ˆbc − U =− ba b ˆac + Uc u 2

(4.21)

c=1 k=1

ˆ is given by the 2-forms The local expression for Q ˆ a = Q(·, ˆ ·, ed , ea ). Q d One notices that the hypothesis (4.5) is rephrased in terms of these forms as X ˆ a. dωba + ωca ∧ ωbc = Q b

(4.22)

(4.23)

c

We may verify proceeding as in the proof of the Lemma 1 and using (4.2) that
ˆ+λ ˆ∧ω+ω∧λ ˆ−λ ˆ∧λ ˆ a. ˆ ·, , · , ed , ea ) = dλ ˆ a := Q( Q d d

(4.24)

ˆ satisfies the zero Combining equations (4.23) and (4.24), one deduces that ω ˆ := ω − λ curvature equation dˆ ω+ω ˆ ∧ω ˆ = 0. (4.25) A suitable version of (2.29) allows us to claim that X X ˆb ˆk − ˆ k ωb + ˆ kλ dU U U a b a b a = 0. b

b

We then prove the following result. 15

(4.26)

Proposition 4 Assume that (4.5) and (4.6) hold. Let M 0 ⊂ M be a connected simply connected open subset. Then there exists an admissible map A ∈ C ∞ (M 0 , SOn+n0 ) so that ˆ A−1 dA = ω − λ

(4.27)

with initial condition A(x0 ) = Id, for a given x0 ∈ M 0 . Proof. We want to assure the existence of an admissible map so that A−1 dA = ω ˆ,

(4.28)

ˆ where ω ˆ = ω − λ. 0 0 If we denote by µ : Mn+n0 R → Rn (n+n ) the projection on the last n0 lines, the condition ˆ (x). The set of admissible maps define a submanifold of (4.14) means that µ(A(x)) = U M × SOn+n0 , namely    ∗ U = (x, A) : A = , (4.29) ˆ (x) U whose tangent space at a point (x, A) is   T(x,A) U = (v, B) : B =

∗ ˆ dU (x) · v

 .

(4.30)

Let ω ¯ ∈ Λ1 (SOn+n0 , son+n0 ) be the Maurer-Cartan form in SOn+n0 . Thus, the equation (4.28) is written as ω ˆ = A∗ ω ¯. (4.31) For solving this equation, we define a 1-form Υ in M 0 × SOn+n0 with values on son+n0 by Υ = π1∗ ω ˆ − π2∗ ω ¯,

(4.32)

where π1 : M × SOn+n0 → M and π2 : M × SOn+n0 → SOn+n0 are the natural projections. We then define the distribution D = ker Υ on U. More precisely (v, B) ∈ D(x,A)

if and only if

ω ˆ x (v) = ω ¯ A (B)

(4.33)

In order to prove that (4.33) defines a distribution we must verify that ker Υ has constant rank. We begin by proving that the differential of π1 restricted to D(x,A) is a monomorphism. In fact, if π1∗ (v, B) = 0 for some (v, B) ∈ D(x,A) then v = 0. Since 0=ω ˆ x (v) = ω ¯ A (B), it follows that B = 0. Therefore, dim ker Υ(x,A) ≤ m. Now, given (v, B) ∈ T(x,A) U we have


µ AΥ(x,A) (v, B) = µ Aˆ ωx (v) − A¯ ωA (B) = µ Aˆ ωx (v) − AA−1 · B

ˆω ˆx (v) = 0 =µ A ω ˆ x (v) − µ B = U ˆ x (v) − dU 16

where in the last equality we used equation (4.26). We then had verified that ImΥ(x,A) ⊂ {B ∈ son+n0 : µ(AB) = 0} Thus, if B ∈ ImΥ(x,A) then B = ω ¯ A (B) for some B tangent to A such that µ(B) = 0. This means that
ImΥ(x,A) ⊂ ω ¯ A ker µA where ker µ
A = {B ∈ TA SOn+n0 : µ(B) = 0}. Since ω ¯ A is an isomorphism, it follows that ω ¯ A ker µA and ker µA have same dimension. Thus, dim ker Υ(x,A) ≥ m. Hence, D(x,A) is m dimensional, for all x ∈ M 0 , A ∈ U. Now we verify the integrability of D. The zero curvature equation (4.25) implies that dΥ = dˆ ω − d¯ ω=ω ˆ ∧ω ˆ −ω ¯ ∧ω ¯ = (¯ ω + Υ) ∧ (¯ ω + Υ) − ω ¯ ∧ω ¯ = ω ¯ ∧Υ+Υ∧ω ¯. Thus if one calculates dΥ at some vector (v, B) ∈ D(x,A) one obtains Υ(v, B) = 0 and then dΥ(v, B) = 0 too. So the ideal ker Υ is differential and then the distribution D is integrable. Since π is a local diffeomorphism between the simply connected domain M 0 and the integral leaf of D passing through (x0 , Id), a standard monodromy reasoning implies that this leaf as the graph x 7→ A(x) of a certain map A ∈ C ∞ (M 0 , SOn+n0 ) which by definition satisfies (4.14) and (4.31).  0

Given an admissible map A : M 0 → SOn+n0 solving (4.27), one defines a frame {ea }m+m a=1 in S along M 0 by (4.14). The corresponding sets of dual 1-forms are related by ˆk

θ =

0 m+m X

Aka ω a .

(4.34)

a=1 0 It stems from (4.1) that the local expression for Jˆk in the frame {ea }n+n a=1 is

X X 1 k 1 n+k ˆl iheb , E ˆr iσ n+k = − u ˆab = − hJˆk ea , eb i = hea , E Ala Arb σlr . lr 2 2 l,r

We then define

l,r

1 X k ˆr 1 XX k r a θˆlk = τlr θ = τ A ω . 2 r 2 a r lr a

In view of these facts, we are able to restate Proposition 1 in the current context.

17

(4.35)

Proposition 5 The admissible frame obtained above as solution of the equation (4.27) satisfies ˆ = A−1 θA, ˆ λ (4.36) 0 where θˆ = (θˆlk )n+n k,l=1 is defined in (4.35).

Proof. It suffices to mimic the proof of Proposition 1 in Section 2.1.1.



We finally define the following 2-forms X

1 XX k r r k r ˆs k r ˆk = 1 Θ τlrk τst + τrs τlt θ ∧ θˆt = τlr τst + τrs τlt Asa Atb ω a ∧ ω b . l 4 s,t 4 s,t

(4.37)

a,b

Then we are able to prove the following result. Proposition 6 The admissible frame defined above as solution of the equation (4.27) satisfies ˆ ˆ = A−1 ΘA, (4.38) Q ˆ = (Θ ˆ k )n+n0 is defined in (4.37). where Θ l k,l=1 Proof. From (4.27) and (4.36) it follows that ˆ = dA−1 ∧ θA ˆ + A−1 dθA ˆ − A−1 θˆ ∧ dA dλ ˆ + A−1 dθA ˆ − A−1 θA ˆ ∧ A−1 dA = −A−1 dA ∧ A−1 θA ˆ ∧λ ˆ + A−1 dθA ˆ −λ ˆ ∧ (ω − λ) ˆ = −(ω − λ) −1 ˆ∧λ ˆ−ω∧λ ˆ−λ ˆ ∧ ω + A dθA. ˆ = 2λ Therefore, in view of (4.24), we conclude that ˆ−λ ˆ∧λ ˆ+ω∧λ ˆ+λ ˆ∧ω =λ ˆ∧λ ˆ + A−1 dθA ˆ ˆ = dλ Q ˆ ∧ A−1 θA ˆ + A−1 dθA ˆ = A−1 θA
= A−1 dθˆ + θˆ ∧ θˆ A.

(4.39)

However, it follows from (4.35), (4.17) and (4.27) that dθˆlk =

1 XX k 1 XX k τlr (dAra ∧ ω a + Ara dω a ) = τlr (dArb ∧ ω b − Ara ωba ∧ ω b ) 2 a r 2 r a,b

=

1 XX k τlr (dArb − Ara ωba ) ∧ ω b 2 r a,b

= −

1 XX k ˆ r τlr (Aλ)b ∧ ω b . 2 r b

18

ˆ = AA−1 θA ˆ = θA. ˆ Hence, one gets However Aλ 1 XX k ˆ r 1 X X k ˆr 1 X k ˆr ˆs τlr (θA)b ∧ ω b = − τlr θs ∧ Asb ω b = − τ θ ∧θ 2 2 2 r,s lr s r b b r,s 1 X k r ˆt ˆs τ τ θ ∧θ . = − 4 r,s,t lr st

dθˆlk = −

On the other hand, one has X r

1 X k r ˆs ˆt τ τ θ ∧θ . θˆrk ∧ θˆlr = 4 r,s,t rs lt

Therefore, one concludes that ˆ dθˆ + θˆ ∧ θˆ = Θ.

(4.40)

Gathering (4.39) and (4.40) we finish the proof.

5



Proof of the Theorem

Part a. In view of the hypothesis in Theorem 1, Proposition 4 implies that there exists an admissible map A : M → SOn+n0 which solves (4.27) and satisfies (4.36) and (4.38) for 0 ˆk n+n0 {θˆk }n+n k=1 and {θl }k,l=1 defined in (4.35) and (4.37), respectively. 0

0

n+n . We then define the We fix in n = Rn+n the orthonormal frame {¯ ek = ωn (Ek )}k=1 following 1-form on M × N with values on n

Π=

∗ πN

ωn −

0 0 n+n X m+m X

k=1

∗ e¯k (Aka ◦ πM )πM ωa,

a=1

where πN : M × N → N and πM : M × N → M are the canonical projections. We then consider the distribution P = ker Π on M × N . Thus, using (4.17) and (4.27), we calculate (omitting projections) X X dΠ = dωn − e¯k dAka ∧ ω a − e¯k Aka dω a a,k

a,k

1 = − [ωn , ωn ] − 2

X

e¯k (Aˆ ω )ka ∧ ω a +

a,k

X

e¯k Aka ωca ∧ ω c

a,c,k

X X X X 1 = − [Π + e¯k θˆk , Π + e¯l θˆl ] − e¯k (Aˆ ω )ka ∧ ω a + e¯k Aka ωca ∧ ω c . 2 k

a,k

l

19

a,c,k

Hence, one has X 1 1 1 X ˆl 1 X ˆk ˆl dΠ = − [Π, Π] − [Π, e¯k θˆk ] − [ e¯l θ , Π] − [¯ ek θ , e¯l θ ] 2 2 2 2 k l k,l X X X ˆ k ∧ ωa + − e¯k (Aω)ka ∧ ω a + e¯k (Aλ) e¯k Aka ωca ∧ ω c . a a,k

a,k

a,c,k

Thus considering equality modulo Π it follows that dΠ = −

X X X 1 X ˆk ˆl ˆc ∧ ωa + θ ∧ θ [¯ ek , e¯l ] − e¯k Akc ωac ∧ ω a + e¯k Akc λ e¯k Akc ωac ∧ ω a a 2 k,l

= −

a,c,k

1X 2

θˆk ∧ θˆl [¯ ek , e¯l ] +

k,l

X

a,c,k

a,c,k

ˆc ∧ ωa. e¯k Akc λ a

a,c,k

However using (4.36) one obtains dΠ = −

XX 1X r ˆk ˆ c (A−1 )b Al ∧ ω a θ ∧ θˆl + e¯k Akc λ e¯r σkl l a b 2 a,b,c k,l

k,l,r

X X 1 XX 1 X k ˆr
ˆl k ˆr = − e¯k σrl θ ∧ θˆl + e¯k θˆlk ∧ θˆl = e¯k θˆlk − σ θ ∧θ . 2 2 r rl r k,l

k,l

k,l

Therefore P is involutive since by (4.35) one has 1 X k ˆr 1 X k ˆr θˆlk = σ θ + µ θ , 2 r rl 2 r lr

(5.1)

l + σ r satisfies µk = µk . This symmetry implies that where µklr = σkr rl lr kl

X l

1 X k ˆr
1 X k ˆr ˆl θˆlk − µlr θ ∧ θ = 0, σrl θ ∧ θl = 2 r 2 l,r

what gives the integrability condition dΠ = 0

mod Π.

We may verify that an integral leaf through the identity y0 in N is a graph over M . The function that graphics this leaf is an isometric immersion f : M → N with initial condition, say, f (x0 ) = y0 , for a given point x0 ∈ M . Indeed, given a tangent vector (v, w) ∈ P(x,y) with y = f (x), we have f∗ (x) · v = w and ωn (w) −

0 0 n+n X m+m X

k=1

e¯k Aka (x)ω a (v) = 0

a=1

20

what yields after left translating both sides by y f∗ (x) · v = w =

0 0 n+n X m+m X

k=1

Ek (f (x))Aka (x)ω a (v).

a=1

Since A(x) is an orthogonal matrix, we conclude that f is an isometric immersion and that ea |f (x) =

0 n+n X

1 ≤ a ≤ m + m0 ,

Ek |f (x) Aka (x),

k=1 0

defines an adapted frame along f with corresponding dual co-frame {ω a }m+m a=1 . Thus, it 0 follows from (4.17) that {ωba }m+m are the connection forms. Thus, (4.27) and (4.36) imply a,b=1 0 n+n k that {θˆ } are the connection forms in N along f with respect to the left-invariant frame l k,l=1 n+n0 {Ek }k=1 . The

0

ˆ k }n+n are the corresponding curvature equation (4.40) assures that {Θ l k,l=1 ˆ forms along f . Finally, (4.37) guarantees that Q is the curvature form in N at points of 0 f (M ) associated to the adapted frame {ea }m+m a=1 . The choice of the initial condition f (x0 ) = y0 is not a serious restriction, since an isometric immersion with initial condition y ∈ N is obtained merely composing f and the left translation by yy0−1 . Part b. From (1.9) and (1.10) it follows that that there exist local orthonormal frames 0 0 respectively adapted to f and f˜ such that the orthogonal matrices and {˜ ea }m+m {ea }m+m a=1 a=1 Aka = hea , Ek i,

A˜ka = h˜ ea , Ek i

(5.2)

satisfy ˜ µ(A) = µ(A).

(5.3)

Moreover, (1.6) and (1.7) imply that the connection forms ω and ω ˜ for adapted frames along f and f˜ satisfy at corresponding points ω = ω ˜. ˜ associated Finally, (5.3), (2.22) and (2.17) imply that the Christoffel tensors λ and λ to these adapted frames are equal at corresponding points. We then conclude that A and A˜ both satisfy the equation A−1 dA = ω − λ (5.4) Now, left translation by f (x0 )f˜(x0 )−1 followed by a suitable rotation in Tf (x0 ) N , if neces˜ 0 ). Hence, the uniqueness of Darboux sary, assure that we may suppose that A(x0 ) = A(x ˜ primitives in a simply connected domain implies that A = A. Thus, we have X ωm |f (x) (f∗ ea ) = e¯k Aka (5.5) k

21

and ωm |f˜(x) (f˜∗ ea ) =

X

e¯k Aka .

(5.6)

k

Therefore, f and f˜ describe integral leaves of the distribution P we defined above passing through the point f (x0 ) ∈ N . The uniqueness part of Frobenius’s theorem implies that f = f˜. This finishes the proof of Theorem 1.

References [1] J. Berndt, Homogeneous hypersurfaces in hyperbolic spaces, Math. Z. 229 (1998), 589-600. [2] J. Berndt, F. Tricerri, L. Vanhecke, Generalized Heisenberg groups and DamekRicci harmonic spaces. Lecture Notes in Mathematics, 1598. Springer-Verlag, Berlin, 1995. [3] P. Ciarlet and F. Larsonneur, On the recovery of a surface with prescribed first and second fundamental forms, J. Math. Pures Appl. 81 (2002), 167-185 [4] E. Damek e F. Ricci, Harmonic analysis on solvable extensions of H-type groups, J. Geom. Anal. 2, 3 (1992), 213-248. [5] M. Dajczer, Submanifolds and isometric immersions. Mathematics Lecture Series, 13. Publish or Perish, Houston, 1990. [6] B. Daniel, Isometric immersions into Sn ×R and Hn ×R and applications to minimal surfaces, to appear in Trans. Amer. Math. Soc. [7] B. Daniel, Isometric immersions into 3-dimensional homogeneous spaces, Comm. Math. Helv. 82, 1 (2007), 87-131. [8] P. Eberlein, Geometry of 2-step nilpotent Lie groups with a left invariant metric, Annales de l’ENS 27, (1994), 611-660. [9] A. Kaplan, Riemannian manifolds attached to Clifford modules, Geom. Dedicata 11, 2 (1981), 127-136. [10] T. Ivey and Landsberg, Cartan for beginners: differential geometry via moving frames and exterior differential systems. Graduate Studies in Mathematics, 61. AMS, Providence, RI, 2003. [11] S. Lodovici, An isometric immersion theorem in Sol3 , Mat. Contem. 30 (2006), 109-124. 22

[12] S. D. Lodovici and F. Manfio, Isometric immersions into a homogeneous lorentzian Heisenberg group and rigidity, preprint, 2008. [13] S. D. Lodovici and P. Piccione, Associated family of G−structure preserving minimal immersions, to appear in Math. Proc. Cambridge Phil. Soc. [14] S. Hervik, Einstein metrics: homogeneous solvmanifolds, generalized Heisenberg spaces and black holes, J. Geom. Phys. 129, 4 (2004), 298-312. [15] F. Manfio, Imers˜ oes isom´etricas em 3-variedades lorentzianas homogˆeneas, Ph. D. Thesis, USP, S˜ao Paulo, 2008. [16] P. Piccione and D. Tausk, An existence theorem for G-structure preserving affine immersions, Indiana Univ. Math. J. 57 (2008), 1431-1465. [17] P. Piccione and D. Tausk, The theory of connections and G-structures, XIV Escola de Geometria Diferencial, Salvador, 2006. [18] M. Spivak, A comprehensive introduction to differential geometry, Publish or Perish, Houston, 2007.

Jorge H. S. de Lira (corresponding author) Departamento de Matem´atica - UFC Campus do Pici, Bloco 914 Fortaleza, Cear´a, Brazil 60455-760 [email protected] Marcos F. de Melo UFC - Campus do Cariri Av. Ten. Raimundo Rocha Juazeiro de Norte, Cear´a, Brazil 60030-200 [email protected]

23