Some classification results on biconservative...

Basic Definitions Biconservative hypersurfaces in pseudo-Euclidean spaces Biconservative Submanifolds in Sn × R Referances

Some classification results on biconservativehypersurfaces in pseudo-Euclidean spaces

Nurettin Cenk TurgayIstanbul Technical University

XIX Geometrical SeminarZlatibor, Serbia

August 28th- September 4th, 2016

Turgay, N. C. Zlatibor 1 / 42


Acknowledgements

In this talk, we would like to present the results obtained in thefollowing recent papers:

• A. Upadhyay and NcT, J. Math. Anal. Appl.(DOI:10.1016/j.jmaa.2016.07.053)

• Y. Fu and NcT, Int. J. Math. (2016).

• F. Manfio, NcT and A. Upadhyay, On biconservativesubmanfolds in Sn × R (pre-print)

114F199

This work was obtained during the TUBITAK 1001 project(Project Name: “Y EUCL2TIP”, Project Number: 114F199).



1 Basic DefinitionsBiharmonic Submanifolds

2 Biconservative hypersurfaces in pseudo-Euclidean spacesBiconservative Hypersurfaces in E4

1

Biconservative Hypersurfaces in E52

3 Biconservative Submanifolds in Sn × RSubmanifolds in Sn × RBiconservative submanifolds with parallel mean curvaturevector



Section 1:

Biharmonic Submanifolds



Biharmonic Mappings

Let φ : M −→ N be a mapping between (Mn, g) and (Nm, 〈, 〉)and τ(φ) = trace∇dφ the tension field of φ.

Biharmonic mappings

If φ is a critical point of the bienergy functional given by

E2(φ) =1

2

∫M|τ(φ)|2vg ,

then it is said to be a a biharmonic map.If, in particular φ = f is an isometric immersion, then M is called abiharmonic submanifold of Nm.



Biconservative Mappings

For a biharmonic map, the bitension field τ2 satisfies the followingassociated Euler-Lagrange equation

τ2(φ) = −∆τ(φ)− traceR(dφ, τ(φ))dφ = 0,

where R is the curvature tensor of N.

Biconservative mappings

Let φ : M −→ N be an isometric immersion satisfying

〈τ2(φ), dφ〉 = 0,

then φ is said to be a biconservative mapping.If, in particular φ = f is an isometric immersion, then M is called abiconservative submanifold of Nm.



Bitension field

The following known splitting result of the bitension field, withrespect to its normal and tangent components, is useful in thestudy of biconservative submanifolds.

Proposition

Let f : Mm → Nn be an isometric immersion between twoRiemannian manifolds. Then f is biharmonic if and only if thetangent and normal components of τ2(f ) vanish, i.e.,

m∇(H2) + 4trA∇⊥· H(·) + 4tr(R(·,H) ·

)T= 0

andtrαf (AH(·), ·)−∆⊥H + 2tr

(R(·,H) ·

)⊥= 0.



Equation of Biconservativity

It follows that an isometric immersion is biconservative if and onlyif

m∇(H2) + 4trA∇⊥· H(·) + 4tr(R(·,H) ·

)T= 0.

• If N is a Riemannian space form Rn(c), then

m∇(H2) + 4trA∇⊥· H(·) = 0.

• If the codimension is 1, i.e., M is a hypersurface, then

S(∇H) = −εnH2∇H.

• If N = Sn × R, then the equation of biconservative becomes

〈H, η〉T = 0.



Section 2:

Biconservative hypersurfaces in

pseudo-Euclidean spaces



Section 2.1: 1

Biconservative Hypersurfaces with

diagonalizable shape operator in E41

1See [Y. Fu and NcT]Turgay, N. C. Zlatibor 10 / 42


Hypersurfaces in E41

It is well-known that the shape operator of a hypersurface in E41

takes one of the following 4 forms for some smooth functionsk1, k2, k3, k4 and ν.

Case I. S =

k1 0 00 k2 00 0 k3

, Case II. S =

k1 1 00 k1 00 0 k3

,

Case III. S =

k1 0 00 k1 1−1 0 k1

, Case IV. S =

k1 −ν 0ν k1 00 0 k3



Shape operator of Biconservative hypersurfaces in E41

Equation of Biconservativity:

S(∇H) = −ε3H

2∇H (BC)

Case I. S =

−ε 3H2 0 00 k2 00 0 k3

, Case II. S =

9H4 1 00 9H

4 00 0 − 3H

2

,

��

��

��XXXXXXXXXXXXXX

Case III. S =

k1 0 00 k1 1−1 0 k1

, Case IV. S =

9H4 −ν 0ν 9H

4 00 0 − 3H

2



Biconservative hypersurfaces with diagonalizable shapeoperator I

Assume that M is a biconservative submanifold and its shapeoperator is diagonalizable. Then, we have

S(∇H) = −ε−3H

2∇H (BC)

Remark

If ∇H = 0, then is satisfied trivially. Thus, we assume that ∇Hdoes not vanish.

Thus, we put e1 = ∇H/|∇H| and let e2, e3 be other principledirections.



Biconservative hypersurfaces with diagonalizable shapeoperator II

Then, we have

e2(k1) = e3(k1) = 0, e1(k1) 6= 0.

Remarks

• It is very easy to observe that multiplicity of k1 is 1.

• If k2 = k3, then do Carmo and Dajczer’s classical result showsthat M is a rotational hypersurfacea.

aSee [Trans. Amer. Math. Soc. 277(1983),685–709]

Hence, we assume that k1 − k2, k1 − k3 and k2 − k3 does notvanish.



Biconservative hypersurfaces with diagonalizable shapeoperator III

By a long computation, we obtain that

TmM = D(m)︸︷︷︸D=span{e2,e3}

⊕ D⊥(m)︸︷︷︸D⊥=span{e1}

and further D and D⊥ are involutive which yields

Proposition

There exists a local coordinate system (s, t, u) such that

e1 =∂

∂s, e2 =

1

E1

∂

∂t, e3 =

1

E2

∂

∂u.



Further,

Proposition

If M has two distinct principal curvature, then it has a localparametrization

x(s, t, u) = φ(s)Θ(t, u) + Γ(s)

for some vector valued functions Θ, Γ and a function φ.If M has three distinct principal curvature, then

x(s, t, u) = φ1(s)Θ1(t) + φ2(s)Θ2(u) + Γ(s)

for some vector valued functions Θ1,Θ2, Γ and functions φ1, φ2.



Biconservative hypersurfaces in E41

We have obtained the following families of biconservativehypersurfaces with diagonalizable shape operator.

Two distinct principal curvatures

• x1(s, t, u) = (f1(s), s cos t sin u, s sin t sin u, s cos u);

• x2(s, t, u) = (ssinhu sin t, scoshu sin t, s cos t, f2(s));

• x3(s, t, u) = (scosht, ssinht sin u, sinht cos u, f3(s));

• x4(s, t, u) =(12s(t2 + u2) + s + f4(s), st, su, 12s(t2 + u2) + f4(s)

).




Zero Gauss-Kronecker Curvature

• A generalized cylinder M20 × E1

1 where M is a biconservativesurface in E3;

• A generalized cylinder M20 × E1 where M is a biconservative

Riemannian surface in E31;

• A generalized cylinder M21 × E1, where M is a biconservative

Lorentzian surface in E31.




Three distinct principal curvatures

• x1(s, t, u) = (scosht, ssinht, f1(s) cos u, f1(s) sin u);

• x2(s, t, u) = (ssinht, scosht, f2(s) cos u, f2(s) sin u);

• A hypersurface in E41 given by

x3(s, t, u) =

(1

2s(t2 + u2) + au2 + s + φ(s), st, (s + 2a)u,

1

2s(t2 + u2) + au2 + φ(s)

), a 6= 0.



Section 2.2:2

Biconservative Hypersurfaces with index 2 in

E52

2See [A. Upadhyay and NcT]Turgay, N. C. Zlatibor 20 / 42


Shape operator of hypersurfaces in E52 I

Let M42 ↪→ E5

2. Then, by choosing an appropriated base field{e1, e2, e3, e4} of the tangent bundle of M, the matrixrepresentation of S can be assumed to be one of the followingforms. Note that in each cases below, g denotes the inducedmetric tensor of M, i.e., gij = 〈ei , ej〉.

• S =

k1 0 0 00 k2 0 00 0 k3 00 0 0 k4

, g =

1 0 0 00 1 0 00 0 −1 00 0 0 −1

;

• S =

k1 1 0 00 k1 0 00 0 k3 00 0 0 k4

, g =

0 −1 0 0−1 0 0 00 0 1 00 0 0 −1

;

• S =

k1 0 1 00 k1 0 00 −1 k1 00 0 0 k4

, g =

0 −1 0 0−1 0 0 00 0 1 00 0 0 −1

;

• S =

k1 0 0 00 k2 0 00 0 k3 β10 0 −β1 k3

, g =

1 0 0 00 −1 0 00 0 1 00 0 0 −1

;



Shape operator of hypersurfaces in E52 II

• S =

k1 1 0 00 k1 0 00 0 k3 10 0 0 k3

, g =

0 −1 0 0−1 0 0 00 0 0 −10 0 −1 0

;

• S =

k1 1 0 00 k1 0 00 0 k3 β10 0 −β1 k3

, g =

0 −1 0 0−1 0 0 00 0 1 00 0 0 −1

;

• S =

k1 β1 0 0−β1 k1 0 00 0 k3 β20 0 −β2 k3

, g =

1 0 0 00 −1 0 00 0 1 00 0 0 −1

;

• S =

k1 β1 1 0−β1 k1 0 10 0 k1 β10 0 −β1 k1

, g =

0 0 −1 00 0 0 1−1 0 0 00 1 0 0

;

• S =

k1 0 1 00 k1 0 00 0 k1 10 1 0 k1

, g =

0 1 0 01 0 0 00 0 0 10 0 1 0

for some smooth functions k1, k2, k3, k4, β1, β2.



Shape operator of biconservative hypersurfaces

Let M be a hypersurface of index 2 in E52 with H as its (first)

mean curvature. If M is biconservative, then its shape operator Shas one of the following forms:

Case I. S =

−2H 0 0 00 k2 0 00 0 k3 00 0 0 k4

, Case II. S =

−2H 0 0 00 k2 −ν 00 ν k2 00 0 0 k4

,

Case III. S =

−2H 0 0 00 k2 1 00 0 k2 00 0 0 k4

, Case IV. S =

−2H 0 0 00 2H 0 00 0 2H −10 1 0 2H

for some smooth functions k1, k2, k3, k4 and ν, where e1 = ∇H‖∇H‖2 .



Classification Results I

Let M be biconservative hypersurface of index 2 in thepseudo-Euclidean space E5

2 and the shape operator S have the form

S = diag(k1, k2, k2, k4), k4 6= k2

Then, M is one of the followings.• x(s, t, u, v) = (φ2sinhv, φ1cosht, φ1sinht cos u, φ1sinht sin u, φ2coshv) ,

• x(s, t, u, v) = (φ2 cos v, φ2 sin v, φ1 cos t, φ1 sin t cos u, φ1 sin t sin u) ,

• x(s, t, u, v) = (φ1cosht sin u, φ1cosht cos u, φ1sinht, φ2 cos v, φ2 sin v) ,

• x(s, t, u, v) = (φ2sinhv, φ1sinht, φ1cosht cos u, φ1cosht sin u, φ2coshv) ,

• x(s, t, u, v) = (φ2coshv, φ1sinht, φ1cosht cos u, φ1cosht sin u, φ2sinhv) ,

• x(s, t, u, v) = (φ1sinhtcosu, φ1sinht sin u, φ1coshu, φ2 cos v, φ2 sin v) ,



Classification Results II

• A hypersurface given by

x(s, t, u, v) =

(s

2

(t2 + u2 − v2

)− av2 + ψ, v(2a + s), st, su,

s

2

(t2 + u2 − v2

)− av2 + ψ − s

) (1)

for a non-zero constants a and a smooth function ψ = ψ(s) such that 1− 2ψ′ < 0;

• A hypersurface given by

x(s, t, u, v) =

s(t2 − u2 − v2

)2

+ av2 + ψ, st, su, v(s − 2a),

s(t2 − u2 − v2

)2

+ av2 + ψ + s

(2)

for a non-zero constants a and a smooth function ψ = ψ(s) such that 1 + 2ψ′ < 0.



A further note

Consider the hypersurface given by

x(s, t1, t2, . . . tn−1) =

(−a1t

21 + a2t

22 + · · · + an−1t

2n−1 +

s‖t‖2

2+ ψ,

t1(s + 2a1), t2(s + 2a2), . . . , tn−1(s + 2a2),

−a1t21 + a2t

22 + · · · + an−1t

2n−1 +

s‖t‖2

2+ ψ − s

),

where ‖t‖2 = t22 + t23 + · · ·+ t2n−1 − t21 .This provides an example of biconservative hypersurface for aparticularly chosen smooth function ψ. Moreover, if all constantsa1, a2, . . . an−1 are distinct, then M has n distinct principalcurvatures.



Section 3:

Biconservative Submanifolds in Sn × R



Section 3.1:

Submanifolds in Sn × R



Basic facts

Given an isometric immersion f : Mm → Sn × R, let ∂t be a unitvector field tangent to the second factor. Then, a tangent vectorfield T on Mm and a normal vector field η along f are defined by

∂t = f∗T + η.

The class AWe will denote by A the class of isometric immersionsf : Mm → Sn × R with the property that T is an eigenvector of allshape operators of f .



The class A

• The class A was introduced in 3 for hypersurfaces

• It extended to submanifolds of Sn × R in 4.

• Trivial examples are• (T = 0) Slices Sn × {t0},• (|T | = 1)The vertical cylinders Nm−1 × R, where Nm−1 is a

submanifold of Sn.

3R. Tojeiro, On a class of hypersurfaces in Sn × R and Hn × R, Bull. Braz.Math. Soc. (N. S.) 41, no. 2, 199–209, (2010).

4B. Mendonca, R. Tojeiro, Umbilical submanifolds of Sn × R, Canad. J.Math. 66, no. 2, 400–428, (2014).



Biconservative Submanifolds

Note that the curvature tensor of Sn × R is

R(X ,Y )Z ={〈Y ,Z 〉X − 〈X ,Z 〉Y }+ (−〈Y ,T 〉+ 〈X ,T 〉)〈Z ,T 〉T+ (〈X ,Z 〉〈Y ,T 〉 − 〈Y ,Z 〉〈X ,T 〉)T .

Therefore, the equation of biconservativity become

m∇‖H‖2 + 4trA∇⊥· H(·) + 4 n〈H, η〉T︸︷︷︸tr(R(·,H)·

)T = 0

for an isometric immersion f : Mm → Sn × R.



Special Cases

m∇‖H‖2 + 4trA∇⊥· H(·) + 4n〈H, η〉T = 0

• If M is a hypersurface, then we have

+S(∇H) +nH

2∇‖H‖+n〈H, η〉T = 0

• If M has parallel mean curvature vector, then 〈H, η〉T = 0.



Section 3.2:

Biconservative submanifolds with parallel

mean curvature vector



Previous works

Biconservative surfaces in Sn × R with parallel mean curvaturestudied in 5.Note that by a Hopf theorem given in 6, M lies S4 × R.

5See D. Fetcu, C. Oniciuc and A. L. Pinheiro, J. Math.Anal.Appl.425(2015), 588–609

6H. Alencar, M. do Carmo and R. Tribuzy, J. Differential Geom. 84 (2010)1–17



Biconservative submanifolds in Sn × RConsider a biconservative submanifolds with parallel non-zero meancurvature vector field and codimension 2. Then, we have

〈H, η〉T = 0.

If T = 0, i.e., ∂t is normal to M. In this case, M is an open part ofthe slice Sn × {t0}. Thus, consider

T 6= 0, 〈H, η〉 = 0.

Note that a simple computation considering 〈H, η〉 = 0 yieldsAH(T ) = 0. Therefore, dimension of the distribution

E0(H) = {X ∈ TM|AH(X ) = 0}

is k and 1 ≤ k < n.



Biconservative submanifolds in Sn × RConsider a biconservative submanifolds with parallel non-zero meancurvature vector field and codimension 2. Then, we have

〈H, η〉T = 0.

If T = 0, i.e., ∂t is normal to M. In this case, M is an open part ofthe slice Sn × {t0}. Thus, consider

T 6= 0, 〈H, η〉 = 0.

Note that a simple computation considering 〈H, η〉 = 0 yieldsAH(T ) = 0. Therefore, dimension of the distribution

E0(H) = {X ∈ TM|AH(X ) = 0}

is k and 1 ≤ k < n.Turgay, N. C. Zlatibor 35 / 42


A further direct computation also yields Aη(T ) ∈ E0(H).Hence, we have

Theorem

Every biconservative submanifolds in Sn × R with codimension 2and parallel mean curvature vector belongs to the class A ifdimE0(H) = 1.

Note that, in particular if M is a surface in Sn × R, then we havedimE0(H) = 1. Hence,

Corollary

Every biconservative surface in Sn × R with parallel meancurvature vector belongs to the class A.

Remark. Compare the result obtained in 7.

7See [Fetcu, Oniciuc and Pinheiro]Turgay, N. C. Zlatibor 36 / 42


The case dimE0(H) = k > 1

Lemma

An isometric immersion f : Mn → Sn+1 × R is biconservative if and onlyif there exists a local orthonormal frame field {e1, e2, . . . , en; en+1, en+2}such that

(1) e1 = T|T | , en+1 = H

|H| , en+2 = η|η| ,

(2) Shape operators along en+1 and en+2 have matrix representationsgiven by

An+1 =

(0 00 S˜1

)and An+2 =

(S˜2 00 B˜

)

for some diagonalized matrices S˜1, B˜ and a symmetric matrix S˜2such that tr(S˜1) = const 6= 0, tr(S˜2) + tr(B˜) = 0 and

tr(B˜S˜1) = 0.

(3) ∇XY ∈ E0(H) whenever X ,Y ∈ E0(H).



An example of biconservative submanifold in S4 × RWe put

Sn(a−2)×R = {(x1, x2, . . . , xn+2) ∈ Rn+2|x21+x22+· · ·+x2n+1 = a2}, n > 1

which implies ∂t = (0, 0, . . . , 0︸︷︷︸(n + 1)-times

, 1).

Example

Let φ = (φ1, φ2, φ3, φ4) : M2 → S2(a−2)× R be an oriented minimalimmersion and a2 + b2 = 1. Then the isometric immersionf : M → S4 × R given by

f (s, u1, u2) =(b cos

s

b, b sin

s

b, φ1(u1, u2), φ2(u1, u2), φ3(u1, u2), φ4(u1, u2)

)is a biconservative immersion with parallel mean curvature vector fieldand dimE0(H) = 2.



Classification Result

Theorem

A biconservative submanifold M3 in S4 × R is either

• (dimE0(H) = 1) belonging to class A or

• (dimE0(H) = 2) congruent to

f (s, u1, u2) =(b cos

s

b, b sin

s

b, φ1(u1, u2), φ2(u1, u2),

φ3(u1, u2), φ4(u1, u2))

described above .



Referances I

J. H. Lira, R. Tojeiro, F. Vitorio, A Bonnet theorem forisometric immersions into products of space forms, (2010).

F. Dillen, J. Fastenakels, J. Van der Veken, Rotationhypersurfaces in S2 × R and H2 × R, (2008).

H. Alencar, M. do Carmo, R. Tribuzy, A Hopf theorem forambient spaces of dimensions higher than three, (2010).B. Daniel, Minimal isometric immersions into S2 × R andH2 × R, (2015).

R. Tojeiro, On a class of hypersurfaces in Sn × R and Hn × R,(2010).

B. Mendonca, R. Tojeiro, Umbilical submanifolds of Sn × R,(2014).



Referances II

N.C. Turgay H-hypersurfaces with 3 distinct principalcurvatures in the Euclidean spaces (accepted 4 days ago)



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