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volume 7, issue 5, article 174, 2006.

Received 29 November, 2005;

accepted 23 November, 2006.

Communicated by:S.S. Dragomir

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Journal of Inequalities in Pure and Applied Mathematics

A NEW OBSTRUCTION TO MINIMAL ISOMETRIC IMMERSIONS INTO A REAL SPACE FORM

TEODOR OPREA

University of Bucharest

Faculty of Maths. and Informatics Str. Academiei 14

010014 Bucharest, Romania.

EMail:teodoroprea@yahoo.com

c

2000Victoria University ISSN (electronic): 1443-5756 348-05

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A New Obstruction to Minimal Isometric Immersions into a

Real Space Form Teodor Oprea

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J. Ineq. Pure and Appl. Math. 7(5) Art. 174, 2006

http://jipam.vu.edu.au

Abstract

In the theory of minimal submanifolds, the following problem is fundamental:

when does a given Riemannian manifold admit (or does not admit) a mini- mal isometric immersion into an Euclidean space of arbitrary dimension? S.S.

Chern, in his monograph [6] Minimal submanifolds in a Riemannian manifold, remarked that the result of Takahashi (the Ricci tensor of a minimal submanifold into a Euclidean space is negative semidefinite) was the only known Rieman- nian obstruction to minimal isometric immersions in Euclidean spaces. A second obstruction was obtained by B.Y. Chen as an immediate application of his fundamental inequality [1]: the scalar curvature and the sectional curvature of a minimal submanifold into a Euclidean space satisfies the inequalityτ≤k.We find a new relation between the Chen invariant, the dimension of the submanifold, the length of the mean curvature vector field and a deviation parameter.

This result implies a new obstruction: the sectional curvature of a minimal sub- manifold into a Euclidean space also satisfies the inequalityk≤ −τ.

2000 Mathematics Subject Classification:53C21, 53C24, 49K35.

Key words: Constrained maximum, Chen’s inequality, Minimal submanifolds.

1. Optimizations on Riemannian Manifolds

Let (N,eg)be a Riemannian manifold, (M, g) a Riemannian submanifold, and f ∈ F(N).To these ingredients we attach the optimum problem

(1.1) min

x∈Mf(x).

The fundamental properties of such programs are given in the papers [7] – [9]. For the interest of this paper we recall below a result obtained in [7].

Theorem 1.1. Ifx₀ ∈M is a solution of the problem(1.1), then i) (gradf)(x₀)∈T_x^⊥

0M, ii) the bilinear form

α:T_x₀M ×T_x₀M →R,

α(X, Y) = Hess_f(X, Y) +eg(h(X, Y),(gradf)(x₀))

is positive semidefinite, where h is the second fundamental form of the submanifoldM inN.

Remark 1. The bilinear formαis nothing else butHessf|M(x₀).

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2. Chen’s Inequality

Let(M, g)be a Riemannian manifold of dimension n, andxa point inM.We consider the orthonormal frame{e1, e2, . . . , en}inTxM.

The scalar curvature atxis defined by

τ = X

1≤i<j≤n

R(e_i, e_j, e_i, e_j).

We denote

δ_M =τ −min(k),

wherekis the sectional curvature at the pointx.The invariantδ_M is called the Chen’s invariant of Riemannian manifold(M, g).

The Chen’s invariant was estimated as the following: “(M, g)is a Rieman- nian submanifold in a real space form Mf(c), varying withcand the length of the mean curvature vector field ofM inMf(c).”

Theorem 2.1. Consider (Mf(c),eg) a real space form of dimension m, M ⊂ Mf(c)a Riemannian submanifold of dimensionn ≥3. The Chen’s invariant of M satisfies

δ_M ≤ n−2 2

n²

n−1kHk²+ (n+ 1)c

,

whereHis the mean curvature vector field of submanifoldMinMf(c). Equality is attained at a point x ∈ M if and only if there is an orthonormal frame {e₁, . . . , e_n} in T_xM and an orthonormal frame {e_n+1, . . . , e_m} in T_x^⊥M in

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which the Weingarten operators take the following form

A_n+1=







hⁿ⁺¹₁₁ 0 0 · · · 0 0 hⁿ⁺¹₂₂ 0 · · · 0 0 0 hⁿ⁺¹₃₃ · · · 0 ... ... ... ... ... 0 0 0 · · · hⁿ⁺¹_nn





 ,

withhⁿ⁺¹₁₁ +hⁿ⁺¹₂₂ =hⁿ⁺¹₃₃ =· · ·=hⁿ⁺¹_nn and

A_r=







h^r₁₁ h^r₁₂ 0 · · · 0 h^r₁₂ −h^r₁₁ 0 · · · 0 0 0 0 · · · 0 ... ... ... ... ... 0 0 0 · · · 0







, r∈n+ 2, m.

Corollary 2.2. If the Riemannian manifold(M, g), of dimensionn ≥3,admits a minimal isometric immersion into a real space formM(c),f then

k≥τ − (n−2)(n+ 1)c

2 .

The aim of this paper is threefold:

• to formulate a new theorem regarding the relation between δ_M, the dimensionn, the length of the mean curvature vector field, and a deviation parametera;

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• to prove this new theorem using the technique of Riemannian program- ming;

• to obtain a new obstruction,k ≤ −τ + ⁽ⁿ²^−n+2)c₂ , for minimal isometric immersions in real space forms.

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3. A New Obstruction To Minimal Isometric Immersions Into A Real Space Form

Let(M, g)be a Riemannian manifold of dimensionn, andaa real number. We define the following invariants

δ_M^a =

( τ−amink, fora≥0, τ−amaxk, fora <0,

whereτ is the scalar curvature, andkis the sectional curvature.

With these ingredients we obtain

Theorem 3.1. For any real numbera ∈[−1,1],the invariantδ^a_M of a Rieman- nian submanifold (M, g), of dimensionn ≥ 3,into a real space formMf(c), of dimensionm,verifies the inequality

δ^a_M ≤ (n²−n−2a)c

2 + n(a+ 1)−3a−1 n(a+ 1)−2a

n²kHk² 2 , whereHis the mean curvature vector field of submanifoldM inMf(c).

If a ∈ (−1,1), equality is attained at the point x ∈ M if and only if there is an orthonormal frame{e₁, . . . , e_n}inT_xM and an orthonormal frame {e_n+1, . . . , e_m}inT_x^⊥M in which the Weingarten operators take the form

A_r =







h^r₁₁ 0 0 · · · 0 0 h^r₂₂ 0 · · · 0 0 0 h^r₃₃ · · · 0 ... ... ... ... ... 0 0 0 · · · h^r_nn





 ,

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with(a+ 1)h^r₁₁ = (a+ 1)h^r₂₂=h^r₃₃=· · ·=h^r_nn,∀r ∈n+ 1, m.

Proof. Consider x ∈ M, {e₁, e₂, . . . , e_n} an orthonormal frame in T_xM, {en+1, en+2, . . . , em}an orthonormal frame inT_x^⊥M anda∈(−1,1).

From Gauss’ equation it follows τ −ak(e₁∧e₂) = (n²−n−2a)c

2 +

m

X

r=n+1

X

1≤i<j≤n

(h^r_iih^r_jj−(h^r_ij)²)−a

m

X

r=n+1

(h^r₁₁h^r₂₂−(h^r₁₂)²).

Using the fact thata∈(−1,1), we obtain (3.1) τ−ak(e₁∧e₂)≤ (n²−n−2a)c

2 +

m

X

r=n+1

X

1≤i<j≤n

h^r_iih^r_jj−a

m

X

r=n+1

h^r₁₁h^r₂₂.

Forr∈n+ 1, m, let us consider the quadratic form f_r :Rⁿ →R, fr(h^r₁₁, h^r₂₂, . . . , h^r_nn) = X

1≤i<j≤n

(h^r_iih^r_jj)−ah^r₁₁h^r₂₂

and the constrained extremum problem maxf_r,

subject toP :h^r₁₁+h^r₂₂+· · ·+h^r_nn =k^r,

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wherek^r is a real constant.

The first three partial derivatives of the functionf_r are

∂f_r

∂h^r₁₁ = X

2≤j≤n

h^r_jj −ah^r₂₂, (3.2)

∂f_r

∂h^r₂₂ = X

j∈1,n\{2}

h^r_jj −ah^r₁₁, (3.3)

∂f_r

∂h^r₃₃ = X

j∈1,n\{3}

h^r_jj. (3.4)

As for a solution (h^r₁₁, h^r₂₂, . . . , h^r_nn) of the problem in question, the vector (grad)(f1)being normal atP, from (3.2) and (3.3) we obtain

n

X

j=1

h^r_jj −h^r₁₁−ah^r₂₂ =

n

X

j=1

h^r_jj −h^r₂₂−ah^r₁₁,

therefore

(3.5) h^r₁₁ =h^r₂₂ =b^r.

From (3.2) and (3.4), it follows

n

X

j=1

h^r_jj −h^r₁₁−ah^r₂₂=

n

X

j=1

h^r_jj −h^r₃₃.

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By using (3.5) we obtainh^r₃₃=b^r(a+ 1). Similarly one gets (3.6) h^r_jj =b^r(a+ 1), ∀j ∈3, n.

Ash^r₁₁+h^r₂₂+· · ·+h^r_nn =k^r,from (3.5) and (3.6) we obtain

(3.7) b^r = k^r

n(a+ 1)−2a . We fix an arbitrary pointp∈P.

The 2-formα :T_pP ×T_pP →Rhas the expression

α(X, Y) = Hess_f_r(X, Y) +hh⁰(X, Y),(grad f_r)(p)i,

where h⁰ is the second fundamental form ofP inRⁿ andh·,·i is the standard inner-product onRⁿ.

In the standard frame ofRⁿ,the Hessian off_rhas the matrix

Hess_f_r =







0 1−a 1 · · · 1 1−a 0 1 · · · 1 1 1 0 · · · 1 ... ... ... ... ... 1 1 1 · · · 0





 .

As P is totally geodesic in Rⁿ, considering a vector X tangent to P at the

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arbitrary pointp, that is, verifying the relationPn

i=1Xⁱ = 0, we have α(X, X) = 2 X

1≤i<j≤n

XⁱX^j−2aX¹X²

=

n

X

i=1

Xⁱ

!2

−

n

X

i=1

(Xⁱ)²−2aX¹X²

=−

n

X

i=1

(Xⁱ)²−a(X¹+X²)²+a(X¹)²+a(X²)²

=−

n

X

i=3

(Xⁱ)²−a(X¹+X²)²−(1−a)(X¹)²−(1−a)(X²)²

≤0.

So Hess_f|M is everywhere negative semidefinite, therefore the point (h^r₁₁, h^r₂₂, . . . , h^r_nn), which satisfies (3.5), (3.6), (3.7) is a global maximum point.

From (3.5) and (3.6), it follows

f_r ≤(b^r)²+ 2b^r(n−2)b^r(a+ 1) +C_n−2² (b^r)²(a+ 1)²−a(b^r)² (3.8)

= (b^r)²

2 [n²(a+ 1)²−n(a+ 1)(5a+ 1) + 6a²+ 2a]

= (b^r)²

2 [n(a+ 1)−3a−1][n(a+ 1)−2a].

By using (3.7) and (3.8), we obtain f_r ≤ (k^r)²

2[n(a+ 1)−2a][n(a+ 1)−3a−1]

(3.9)

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= n²(H^r)²

2 · n(a+ 1)−3a−1 n(a+ 1)−2a . The relations (3.1) and (3.9) imply

(3.10) τ−ak(e₁∧e₂)≤ (n²−n−2a)c

2 +n(a+ 1)−3a−1

n(a+ 1)−2a · n²kHk² 2 . In (3.10) we have equality if and only if the same thing occurs in the inequality (3.1) and, in addition, (3.5) and (3.6) occur. Therefore

(3.11) h^r_ij = 0, ∀r ∈n+ 1, m, ∀i, j ∈1, n, with i6=j and

(3.12) (a+ 1)h^r₁₁= (a+ 1)h^r₂₂ =h^r₃₃=· · ·=h^r_nn,∀r ∈n+ 1, m.

The relations (3.10), (3.11) and (3.12) imply the conclusion of the theorem.

Remark 2.

i) Making a to converge at 1 in the previous inequality, we obtain Chen’s Inequality. The conditions for which we have equality are obtained in [1]

and [7].

ii) Fora= 0we obtain the well-known inequality τ ≤ n(n−1)

2 (kHk²+c).

The equality is attained at the point x ∈ M if and only if x is a totally umbilical point.

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iii) Makingaconverge at−1in the previous inequality, we obtain

δ_M⁻¹ ≤ (n²−n+ 2)c

2 +n²kHk² 2 .

The equality is attained at the pointx ∈ M if and only if there is an or- thonormal frame {e₁, . . . , e_n} in T_xM and an orthonormal frame {e_n+1, . . . , e_m} inT_x^⊥M in which the Weingarten operators take the fol- lowing form

A_r=







h^r₁₁ 0 0 · · · 0 0 h^r₂₂ 0 · · · 0 0 0 0 · · · 0 ... ... ... ... ... 0 0 0 · · · 0





 ,

withh^r₁₁ =h^r₂₂,∀r∈n+ 1, m.

Corollary 3.2. If the Riemannian manifold(M, g), of dimensionn ≥3,admits a minimal isometric immersion into a real space formM(c),f then

τ − (n−2)(n+ 1)c

2 ≤k ≤ −τ +(n²−n+ 2)c

2 .

Corollary 3.3. If the Riemannian manifold(M, g), of dimensionn ≥3,admits a minimal isometric immersion into a Euclidean space, then

τ ≤k ≤ −τ.

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References

[1] B.Y. CHEN, Some pinching classification theorems for minimal submani- folds, Arch. Math., 60 (1993), 568–578.

[2] B.Y. CHEN, A Riemannian invariant for submanifolds in space forms and its applications, Geom. Topology of Submanifolds, World Scientific, Leu- ven, Brussel VI (1993), 58–81.

[3] B.Y. CHEN, A Riemannian invariant and its applications to submanifolds theory, Results in Mathematics, 27 (1995), 17–26.

[4] B.Y. CHEN, Mean curvature and shape operator of isometric immersions in real-space-forms, Glasgow Math. J., 38 (1996), 87–97.

[5] B.Y. CHEN, Some new obstructions to minimal Lagrangian isometric im- mersions, Japan. J. Math., 26 (2000), 105–127.

[6] S.S. CHERN, Minimal Submanifolds in a Riemannian Manifold, Univ. of Kansas, Lawrence, Kansas, 1968.

[7] T. OPREA, Optimizations on Riemannian submanifolds, An. Univ. Buc., LIV(1) (2005), 127–136.

[8] C. UDRI ¸STE, Convex functions and optimization methods on Riemannian manifolds, Mathematics its Applications, 297, Kluwer Academic Publishers Group, Dordrecht, 1994.

[9] C. UDRI ¸STE, O. DOGARU AND I ¸TEVY, Extrema with nonholonomic constraints, Geometry, Balkan Press, Bucharest, 2002.