ON HYPERSURFACES IN A LOCALLY AFFINE RIEMANNIAN BANACH MANIFOLD II

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ON HYPERSURFACES IN A LOCALLY AFFINE RIEMANNIAN BANACH MANIFOLD II

EL-SAID R. LASHIN and TAREK F. MERSAL Received 8 March 2002

In our previous work (2002), we proved that an essential second-order hypersurface in an infinite-dimensional locally affine Riemannian Banach manifold is a Riemannian manifold of constant nonzero curvature. In this note, we prove the converse; in other words, we prove that a hypersurface of constant nonzero Riemannian curvature in a locally affine (flat) semi- Riemannian Banach space is an essential hypersurface of second order.

2000 Mathematics Subject Classification: 53C20, 53C40.

1. Introduction. LetMbe an infinite-dimensional Banach manifold of classC^k,k≥1, modelled on a Banach spaceE, and letg¹¯be a symmetric bilinear form defined onM, that is,

1

g¯∈L2(M;R). The metricg¹¯is said to be strongly nonsingular if

1

g¯associates a mappingg¹¯

∗

:x∈M→g¹¯

∗ x=g¹¯

∗

(x,·)∈L(M;R)which is bijective [2]. Let

1

¯Γ be the linear connection onM. AC^kBanach manifold(M,

1

¯Γ),k≥3, is called locally affine if its curvature and torsion tensors are zero. In general, it is proved in [2] that a Banach manifold(M,

1

¯Γ)is locally affine if and only if there exists an atlasᏭonMsuch that for any chartc∈Ꮽ,Γ¹≡0, whereΓ¹is the model of the linear connection

1

¯Γ. The hypersurface N⊂Mwhich is defined by the equationg¹¯_x(x,⁻ x)⁻ =er²,e= ±1, 0≠r∈R, is called an essential hypersurface of the second order in the spaceM(see [2]).

2. Hypersurface of nonzero constant Riemannian curvature in a locally affine Banach manifold. LetM be a locally affine Banach manifold and assume thatg¹¯is a strongly nonsingular metric onM, then the pair(M,g)¹¯ is a Riemannian Banach mani- fold. Denote byi: ¯x∈N→i(¯x)=x¯∈Mthe inclusion mapping. Letc=(U ,Φ, E)be a chart at ¯x∈Mand letd=(V ,Ψ, F⊆E)be a chart at ¯x∈N, where the Banach spaces E andF are the models of the manifoldsM and Nwith respect to the charts c, and d, respectively. Furthermore, we have thatΨ(¯x)=xis the model of the point ¯x with respect to the chartd, z=Φ(¯x)is the model of ¯xwith respect to the chartc, andiis the model ofiwith respect to the chartscandd. Then we have an inclusion

i:x=Ψ(x)¯ ∈Ψ(V )⊂F →i(x)=z=Φ(¯x)∈Φ(V )⊂E (2.1)

of a hypersurface of a semi-Riemannian Banach spaceE.

In this case, (2.1) is called the local equation of the submanifoldN⊂Mwith respect to the charts cand d. AlsoN will be a Riemannian submanifold ofM with induced metricg, which is defined by the rule²¯

2

¯

g_xX¯1,X¯2

=g¹¯_i(x) TxiX¯1

, TxiX¯2

, (2.2)

for all ¯x∈Nand ¯X1,X¯2∈Tx¯N, whereTx¯i:Tx¯N→Tx¯Mis the tangent mapping ofiat the point ¯x∈N(see [1]).

Assume thatg²¯is a strongly nonsingular metric onN. Also we have thatMandNare Riemannian manifolds with free-torsion connections

1

¯Γ and

2

¯Γ, respectively, such that

1

∇¯g¹¯=0 and

2

∇¯g²¯=0 (see [3, 4]). LetX1, X2∈F be the models of ¯X1, ¯X2∈Tx¯N with respect to the chartdonN. ThenY1=Dix(X1)andY2=Dix(X2)are the models of ¯X1

and ¯X2with respect to the chartconM.

In this case, the local equation of (2.2) takes the form g2_x

X1, X2

=g¹_x Dix

X1

, Dix

X2

. (2.3)

Theorem2.1. A local hypersurface of constant nonzero Riemannian curvature in a locally affine (flat) semi-Riemannian Banach space is an essential hypersurface of second order.

Proof. LetNbe a local hypersurface of constant curvatureK0of the Banach type in the Riemannian manifold(M,g)¹¯ such that dimN >2. We know that the first differential equation of the hypersurfaceN⊂Mhas the form (see [5])

∇2Dix(X, Y )=eAx(X, Y )ξx, (2.4)

where ¯ξx∈T₀¹₊⁺₀⁰(M)=T₀¹(M)is a unit vector inMorthogonal toNat the point ¯x∈M, that is,

1

¯ gξ¯x,ξ¯x

=e, g¹¯ξ¯x,X¯

=0, (2.5)

for all ¯x∈N⊂M and all ¯X∈TxN, andAx is the second fundamental form for the hypersurfaceNwhich is defined by the equality (see [5])

Ax(X, Y )=g¹_x

D²ix(X, Y ), ξx

= −g¹_x

Dix(X), Dξx(Y )

. (2.6)

Taking into account thatTxi∈T₀₊₁¹⁺⁰(N)is a mixed tensor of type(1+0,0+1)on the submanifold N(see [7]), ¯ξx∈T₀¹(M), and (2.6), we conclude that Ax is a symmetric tensor of type(0,2)onNat the point ¯x∈N.

Now letξ:x=Ψ(¯x)∈Ψ(V )⊂F→ξx∈Ebe the model of the vector field

ξ¯: ¯x∈N →ξ¯x¯∈Tx¯M, (2.7)

with respect to the chartscanddat the point ¯x. Then the local equations of equalities (2.5) take the form

g1 ξx, ξx

=e, g¹

Dix(X), ξx

=0, (2.8)

for allx∈Ψ(V )⊂F and allX∈F. Furthermore, the integral condition for (2.4) takes the form

g1 Dix

2

Rx(Y;Z, X), Dix(S)

=g²_x2

Rx(Y;Z, X), S

=eAx Z, Y

Ax X, S

. (2.9)

Remark2.2. In formula (2.9), there exists an alternation with respect to the under- lined vectors without division by 2. This convention will be used henceforth.

Similarly, the second differential equation of the hypersurfaceN⊂Mwill be (see [5])

Dξx(X)=Dix Hx(X)

, (2.10)

whereHx∈L(F;F ). Also by using (2.6), we find that

Ax(X, Y )= −g¹_x

Dix(X), Dξx(Y )

= −g¹_x

Dix(X), Dix

Hx(Y )

= −g²_x

X, Hx(Y ) , (2.11)

that is,

g2_x

X, Hx(Y )

= −Ax(X, Y ), (2.12)

for allx=Ψ(¯x)∈Ψ(V )⊂F and allX, Y ∈F. Furthermore, the integral condition for (2.10) has the form (see [5])

∇2Ax

X;Z, Y

=0, (2.13)

for allx=Ψ(¯x)∈Ψ(V )⊂Fand allX, Y , Z∈F. Now we find that

g2_x2

Rx(Y;Z, X), S

=g¹_x Dix

2

Rx(Y;Z, X)

, Dix(S)

=eAx

Z, Y Ax

X, S

. (2.14)

SinceNis a hypersurface of constant curvature, then (2.14) takes the form (see [2]) g2_x

K0

g2_x(Z, Y )X, S

=eAx Z, Y

Ax X, S

, (2.15)

whereK0∈Ris a constant independent of the choice of the point, and is called the curvature of the hypersurfaceN. Then, we obtain

Ax(Z, Y )Ax(X, S)−Ax(X, Y )Ax(Z, S)

=K₂

g_x(Z, Y )g²_x(X, S)−g²_x(X, Y )g²_x(Z, S)

, (2.16)

for allx=Ψ(¯x)∈Ψ(V )⊂Fand allX, Y , Z, S∈F, whereK=K0/e.

Now we prove thatAx is a weakly nonsingular form. LetX be a fixed vector and Ax(X, Y )=0, for allY∈F. Then, from (2.16) we obtain

g2_x(Z, Y )g²_x(X, S)−g²_x(X, Y )g²_x(Z, S)=0, (2.17)

for allY ∈F, that is, g²_x(Y ,g²_x(X, S)·Z−g²_x(Z, S)·X)=0. By using thatg²_x is non- singular, we obtaing²_x(X, S)·Z−g²_x(Z, S)·X=0, for allx=Ψ(¯x)∈Ψ(V )⊂F and all X, Z, S∈F. Since dimE >2, then, for anyS, we can chooseZwhich is not a multiple ofXand thusg²_x(X, S)=0, for allS∈F. Butg²_xis nonsingular, hence,X=0 and this proves thatAxis a weakly nonsingular form.

Now from (2.12) and (2.16), we obtain g2_x

Z, Hx(Y )² g_x

X, Hx(S)

=K2

g_x Z, Y²

g_x X, S

, (2.18)

and then we have g2_x

Z,g²_x

X, Hx(S)

·Hx(Y )−g²_x

X, Hx(Y )

·Hx(S)

−K₂

g_x(X, S)·Y−g²_x(X, Y )·S

=0, ∀Z∈F .

(2.19)

Taking into account that the metric tensorg²_xis nonsingular, we obtain g2_x

X, Hx(S)

·Hx(Y )−g²_x

X, Hx(Y )

·Hx(S)

−Kg²_x(X, S)·Y+Kg²_x(X, Y )·S=0.

(2.20)

Furthermore, we find g2_x

X, Hx(Y )

=Ax(X, Y )=Ax(Y , X)=g²_x

Y , Hx(X)

=g²_x

Hx(X), Y

, (2.21)

that is,

g2_x

X, Hx(Y )

=g²_x

Hx(X), Y

, (2.22)

and then from (2.20) and (2.22), we obtain g2_x

Hx(X), S

·Hx(Y )−g²_x

Hx(X), Y

·Hx(S)

−Kg²_x(X, S)·Y+Kg²_x(X, Y )·S=0,

(2.23)

for allx=Ψ(¯x)∈Ψ(V )⊂Fand allX, Y , S∈F.

Since dimF >2, then, for everyX, Y∈Fsuch thatg²_x(X, Y )=0, there exists a vector S ∈F orthogonal to eachX and Hx(X)[2]. Using this fact in (2.23) and taking into account (2.12), we obtainAx(X, Y )·Hx(S)=0. By using the nonsingularity of the tensor Ax, we conclude thatAx(X, Y )=0. Since, for any pair of vectorsX, Y∈F,g²_x(X, Y )=0 implies thatAx(X, Y )=0, then there exists a real numberλsuch that (see [2])

Ax(X, Y )=λg²_x(X, Y ). (2.24) Substituting (2.24) into (2.16), we obtain

λ^{2 2}g_x Z, Y²

g_x X, S

=Kg²_x Z, Y²

g_x X, S

, (2.25)

for allx=Ψ(¯x)∈Ψ(V )⊂Fand allX, Y , Z, S∈F. Taking into account the nonsingularity ofg²_x, we obtainλ²=K=K0/e. It is convenient to putK0=e/r², whereris a nonzero real number ande= ±1, then we haveλ= ±1/r. We find that in our case, it is convenient to takeλ= −1/r. Substitutingλin (2.24), we obtain

Ax(X, Y )= −1 r

g2_x(X, Y ), (2.26)

and in fact this equation is the unique solution, up to sign, of (2.9) and (2.13). Substi- tuting this solution in (2.12), we have

g2_x

X, Hx(Y )

= 1 r

g2_x(X, Y ), ∀x∈Ψ(V )⊂F ,∀X, Y∈F , (2.27)

which implies thatHx(Y )=(1/r )Y. Hence (2.10) will be Dξx(X)=1

rDix(X). (2.28)

Integrating this equation gives usξx=(1/r )i(x). Then g1

i(x), i(x)

=r^{2 1}g ξx, ξx

. (2.29)

Lettingy=i(x)and using equalities (2.8), the above equation takes the form

g(y, y)1 =er², ∀x∈Ψ(V )⊂F , e= ±1. (2.30)

This last equation shows that the hypersurfaceN⊂Mof constant nonzero Riemannian curvature will be locally an essential hypersurface of second order, and this completes the proof.

References

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Linear Connection, Kazan. Gos. Univ., Kazan, 1983.

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[4] ,Foundations of Differential Geometry. Vol. II, John Wiley & Sons, New York, 1969.

[5] E. R. Lashin,On hypersurfaces in Banach manifolds, J. Faculty of Education, Ain-Shams Univ.

20(1995), 257–267.

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El-Said R. Lashin: Department of Mathematics, Faculty of Science, Menoufiya University, Menoufiya 32511, Egypt

E-mail address:[email protected]

Tarek F. Mersal: Department of Mathematics, Faculty of Science, Menoufiya University, Menoufiya 32511, Egypt

E-mail address:[email protected]