Real hypersurfaces of type (A) in complex two-plane Grassmannians related to the commuting shape operator

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DOI 10.1515 / forum-2012-0029 © de Gruyter 2013

Real hypersurfaces of type (A) in complex two-plane Grassmannians related to the commuting shape operator

Imsoon Jeong and Young Jin Suh

Communicated by Karl Strambach

Abstract.We give some characterizations of real hypersurfaces of type (A) in complex two plane GrassmanniansG₂.C^mC2/, that is, a tube over a totally geodesicG₂.C^mC1/in G₂.C^mC2/with the commuting conditionA₁ D₁Afor the shape operatorA, the structure tensorsand₁, together with additional geometric conditions.

Keywords.Real hypersurfaces, complex two-plane Grassmannians, Hopf hypersurface, commuting shape operator.

2010 Mathematics Subject Classification.Primary 53C40; secondary 53C15.

Introduction

We denote byG2.C^m^C²/the set of all complex two-dimensional linear subspaces in C^m^C². This Riemannian symmetric spaceG2.C^m^C²/has a remarkable geometric structure. It is the unique compact irreducible Riemannian manifold with both a Kähler structureJ and a quaternionic Kähler structureJ not containingJ. Namely,G2.C^m^C²/is a unique compact, irreducible, Kähler, quaternionic Kähler manifold which is not a hyper-Kähler manifold.

The almost contact structure vector field defined by D JN is said to be a Reeb vector field, where N denotes a local unit normal vector field of M in G2.C^m^C²/. The almost contact 3-structure vector fields ¹1; 2; 3º for the 3-dimensional distributionD^?ofM inG2.C^m^C²/are defined by D JN . D 1; 2; 3), whereJ denotes a canonical local basis of a quaternionic Kähler structureJ and the tangent space is decomposed asTxM DD˚D^?,x2M.

Accordingly, inG2.C^m^C²/we can consider the natural geometric condition for a real hypersurfaceM that the1-dimensional distributionŒDSpan¹ºand the

This work was supported by grant no. NRF-2011-220-C00002 from National Research Foundation of Korea. The first author was supported by grant no. BSRP-2012-0004248. The second author was supported by grant no. BSRP-2012-0007402.

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3-dimensional distributionD^? D Span¹1; 2; 3ºare invariant under the shape operatorAofM (see [2–4]). Then by using a result of Alekseevskii [1], Berndt and Suh [3] proved the following:

Theorem A([3]).LetM be a connected orientable real hypersurface in the space G2.C^m^C²/,m3. Then bothŒandD^?are invariant under the shape operator ofM if and only if

(A) M is an open part of a tube around a totally geodesic G2.C^m^C¹/ in G2.C^m^C²/, or

(B) mis even, saym D 2n, andM is an open part of a tube around a totally geodesicHPⁿinG2.C^m^C²/.

The Reeb vector field is said to be Hopf if it is invariant under the shape operator A. The 1-dimensional foliation of M by the integral manifolds of the Reeb vector field is said to be a Hopf foliation of M. We say that M is a Hopf hypersurfaceinG2.C^m^C²/if and only if the Hopf foliation ofM is totally geodesic. By the formulas in Section 2 it can be easily checked thatM is Hopf if and only if the Reeb vector fieldis Hopf.

On the other hand, we say that the Reeb flow onM inG2.C^m^C²/isisometric, when the Reeb vector fieldonM is Killing. In [4], Berndt and Suh gave some equivalent conditions for isometric Reeb flow. Among them, we want to consider a commuting condition between the shape operatorAand the structure tensor, that is,A DA.

By such a commuting condition, a characterization of real hypersurfaces of type (A) in Theorem A was given in terms of the Reeb flow onMas follows:

Theorem B([4]).LetM be a connected orientable real hypersurface in the space G2.C^m^C²/,m3. Then the Reeb flow onM is isometric if and only ifM is an open part of a tube around a totally geodesicG2.C^m^C¹/inG2.C^m^C²/.

In [8], Suh considered the condition that the almost contact 3-structure tensors ¹1; 2; 3º commute with the shape operator Aof a real hypersurface M in G2.C^m^C²/, and he proved that there does not exist any real hypersurface M in G2.C^m^C²/with AX D AX, D 1; 2; 3, for any tangent vector field X onM. In addition, he gave a characterization of a real hypersurface M of type (B) under the assumption thatM is a Hopf hypersurface inG2.C^m^C²/with AX DAX,D1; 2; 3, for any tangent vector fieldX onT0. Here, the dis- tributionT0is defined byT0 D ¹X 2TM j?Xº, andTM denotes the tangent bundle ofM inG2.C^m^C²/(see [8]).

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On the other hand, from the geometric structure ofG2.C^m^C²/in Section 1 we have a natural commuting condition thatJJDJJ,D1; 2; 3, between Kähler structureJ and quaternionic Kähler structureJ. In the caseD1, the commuting conditionJJ1DJ1J gives

1X D1XC.X /1 1.X /

for any vector fieldX tangent toM inG2.C^m^C²/, where; 1W TM!TM denote the structure operators associating to a tangent vectorX 2TMthe tangential component ofJ X; J1X 2TM, respectively. WhenJ12Jsuch thatJ1N DJN in Proposition E of Section 3, then the Reeb vector field satisfies D 1and the formula becomes1D1.

Summing up all the statements mentioned above, naturally we ask what we can say about commuting between the shape operatorAand the local tensor1com- posed by two structure tensors1and. Accordingly, in this paper we consider a new condition that the shape operatorAcommutes with the tensor1 for a real hypersurfaceM inG2.C^m^C²/as follows:

A1X D1AX ()

for any tangent vector fieldXonM.

The geometric meaning of the above condition () is that any eigenspaces of the shape operator are invariant under the composite transformation1ofTM.

Now in this paper we want to give complete classifications of real hypersurfaces M inG2.C^m^C²/under the assumption of a commuting shape operator () and some related conditions.

First, let us give a classification forD^?-invariant hypersurfaces inG2.C^m^C²/ as follows:

Theorem 1.LetM be a connected orientableD^?-invariant real hypersurface in G2.C^m^C²/,m 3. If the shape operator A satisfies the commuting condition (), thenM is an open part of a tube around a totally geodesic G2.C^m^C¹/ in G2.C^m^C²/.

Second, for hypersurfaces inG2.C^m^C²/with the Reeb vector field2D^?we give a classification as follows:

Theorem 2.LetM be a connected orientable real hypersurface inG2.C^m^C²/, m 3. If the shape operatorA satisfies the commuting condition () and the Reeb vector field belongs to the distributionD^?, thenM is an open part of a tube around a totally geodesicG2.C^m^C¹/inG2.C^m^C²/.

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Third, for hypersurfaces inG2.C^m^C²/with constant principal curvature of the Reeb vector field we give a classification as follows:

Theorem 3.LetM be a connected orientable real hypersurface inG2.C^m^C²/, m 3. If the shape operatorA satisfies the commuting condition () and the principal value of the Reeb vector fieldis constant, thenM is an open part of a tube around a totally geodesicG2.C^m^C¹/inG2.C^m^C²/.

Finally, together with another commuting condition related to the normal Jacobi operator, that is,RNN ıDı NRN, we have the following:

Theorem 4.LetM be a connected orientable real hypersurface inG2.C^m^C²/, m 3. If the shape operatorA satisfies the commuting condition () and the normal Jacobi operator commutes with the structure tensor, thenM is an open part of a tube around a totally geodesicG2.C^m^C¹/inG2.C^m^C²/.

1 Riemannian geometry ofG₂.C^mC2/

In this section we summarize some basic material about G2.C^m^C²/. For de- tails we refer to [2–4]. By G2.C^m^C²/ we denote the set of all complex two- dimensional linear subspaces inC^m^C². The special unitary groupGDS U.mC2/

acts transitively on G2.C^m^C²/ with stabilizer isomorphic to K D S.U.2/

U.m// G. Then G2.C^m^C²/can be identified with the homogeneous space G=K, which we equip with the unique analytic structure for which the natural ac- tion of G onG2.C^m^C²/ becomes analytic. Denote bygandk the Lie algebra of G andK, respectively, and bymthe orthogonal complement of k in gwith respect to the Cartan–Killing formBofg. ThengDk˚mis an Ad.K/-invariant reductive decomposition of g. We puto D eK and identifyToG2.C^m^C²/with min the usual manner. SinceB is negative definite ong, its negative restricted tommyields a positive definite inner product onm. By Ad.K/-invariance of B this inner product can be extended to a G-invariant Riemannian metricg on G2.C^m^C²/. In this wayG2.C^m^C²/becomes a Riemannian homogeneous space, even a Riemannian symmetric space. For computational reasons we normalizeg such that the maximal sectional curvature of.G2.C^m^C²/; g/is eight.

WhenmD1,G2.C³/is isometric to the two-dimensional complex projective spaceCP²with constant holomorphic sectional curvature eight.

WhenmD2, we note that the isomorphism Spin.6/'S U.4/yields an isom- etry betweenG2.C⁴/and the real Grassmann manifoldG₂^C.R⁶/of oriented two- dimensional linear subspaces inR⁶. In this paper, we will assumem3.

The Lie algebra k has the direct sum decomposition k D su.m/˚su.2/˚ R, where R denotes the center of k. Viewing k as the holonomy algebra of

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G2.C^m^C²/, the centerRinduces a Kähler structureJ and thesu.2/-part a quaternionic Kähler structureJ onG2.C^m^C²/. IfJis any almost Hermitian structure inJ, thenJJ DJJ, andJJ is a symmetric endomorphism with.JJ/²DI and tr.JJ/D0forD1; 2; 3.

A canonical local basis¹J1; J2; J3ºofJconsists of three local almost Hermit- ian structuresJ inJ such thatJJC1 D JC2 D JC1J, where the index is taken modulo three. SinceJ is parallel with respect to the Riemannian con- nectionrN of.G2.C^m^C²/; g/, there exist three local one-formsq1; q2; q3for any canonical local basis¹J1; J2; J3ºofJ such that

rNXJ DqC2.X /JC1 qC1.X /JC2 (1.1) for all vector fieldsX onG2.C^m^C²/.

The Riemannian curvature tensorRQofG2.C^m^C²/is locally given by R.X; Y /ZQ Dg.Y; Z/X g.X; Z/Y Cg.J Y; Z/J X

g.J X; Z/J Y 2g.J X; Y /J Z

C

3

X

D1

®g.JY; Z/JX g.JX; Z/JY 2g.JX; Y /JZ¯

C

3

X

D1

®g.JJ Y; Z/JJ X g.JJ X; Z/JJ Y¯

; (1.2)

where¹J1; J2; J3ºdenotes a canonical local basis ofJ.

2 Some fundamental formulas

In this section we derive some basic formulas and the Codazzi equation for a real hypersurface inG2.C^m^C²/(see [5, 7–10]).

Let M be a real hypersurface of G2.C^m^C²/, that is, a hypersurface of G2.C^m^C²/with real codimension one. The induced Riemannian metric on M will also be denoted by g, andr denotes the Riemannian connection of.M; g/.

LetN be a local unit normal vector field ofM andAthe shape operator of M with respect toN.

Now let us put

J XDXC.X /N; JXDXC.X /N (2.1) for any tangent vector fieldX of a real hypersurfaceM inG2.C^m^C²/, whereN denotes a unit normal vector field ofM inG2.C^m^C²/. From the Kähler structure

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J ofG2.C^m^C²/there exists an almost contact metric structure.; ; ; g/induced onM in such a way that

²X D XC.X /; ./D1; D0; .X /Dg.X; / (2.2) for any vector field X onM. Furthermore, let¹J1; J2; J3º be a canonical local basis of J. Then the quaternionic Kähler structureJ of G2.C^m^C²/, together with the conditionJJC1 DJC2 D JC1J in Section1, induces an almost contact metric 3-structure.; ; ; g/onM as follows:

²X D XC.X /; ./D1; D0;

C1 D C2; C1DC2; C1X DC2XCC1.X /; C1X D C2XC.X /C1

for any vector fieldX tangent toM. Moreover, from the commuting property of JJ D JJ, D 1; 2; 3, in Section 1 and (2.1), the relation between these two contact metric structures.; ; ; g/and.; ; ; g/,D1; 2; 3, can be given by

X DXC.X / .X /; .X /D.X /; D:

On the other hand, from the Kähler structureJ, that is,rNJ D0and the quaternionic Kähler structure J (see (1.1)), together with the Gauss and Weingarten formulas, it follows that

.rX/Y D.Y /AX g.AX; Y /; rX DAX;

rX DqC2.X /C1 qC1.X /C2CAX;

.rX/Y D qC1.X /C2Y CqC2.X /C1Y C.Y /AX g.AX; Y /:

Summing up these formulas, we find the following:

rX./D rX./

D.rX/C.rX/

DqC2.X /C1 qC1.X /C2CAX g.AX; /C./AX:

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Using the above expression (1.2) for the curvature tensorRQ ofG2.C^m^C²/, the equation of Codazzi becomes

.rXA/Y .rYA/X D.X /Y .Y /X 2g.X; Y /

C

3

X

D1

®.X /Y .Y /X 2g.X; Y /¯

C

3

X

D1

®.X /Y .Y /X¯

C

3

X

D1

®.X /.Y / .Y /.X /¯ :

3 Proof of Theorems 1 and 2

Now let us assume thatM is a real hypersurface inG2.C^m^C²/with commuting shape operator, that is, the shape operatorAofM commutes with the local tensor 1as in () for any tangent vector fieldX onM.

First of all, we give the following key lemma.

Lemma 5.LetM be a real hypersurface inG2.C^m^C²/,m 3. IfM has the commuting shape operator(), thenMis Hopf.

Proof. By using the assumption () forX D, we have 0DA1D1A:

From this, by applying1we have

0D AC1.A/1: And by applyingin the above formula we obtain

AD.A/ 1.A/1: (3.1)

Next by using the assumption () forX D1and applying1we have 1AC1./1A1D A1C1.A1/1: From this, by taking an inner product with, we obtain

g.A; 1/D0: (3.2)

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Thus by using (3.1) and (3.2), we have A D˛;

where˛D.A/. So we complete the proof of the lemma.

Summing up Lemma 5 and Theorem A, we know that any connected orientable D^?-invariant real hypersurface inG2.C^m^C²/,m3, satisfying the commuting shape operator condition () is congruent to either type (A) or (B).

Before giving the proof of Theorem 1, let us check whether or not the shape operatorAfor real hypersurfaces of type (A) in Theorem A satisfies condition ().

So we recall the following proposition due to Berndt and Suh [3] concerned with a tube of type (A).

Proposition E([3]).LetM be a connected real hypersurface ofG2.C^m^C²/. Sup- pose thatADD,A D˛, andis tangent toD^?. LetJ12J be the almost Hermitian structure such thatJN DJ1N. ThenM has three (ifr D=2p

8) or four (otherwise) distinct constant principal curvatures

˛Dp 8cot.p

8r/; ˇ Dp 2cot.p

2r/; D p 2tan.p

2r/; D0

with somer 2.0; =p

8/. The corresponding multiplicities are m.˛/D1; m.ˇ/D2; m./D2m 2Dm./;

and the corresponding eigenspaces are

T˛ DR DRJN DR1DSpan¹º DSpan¹1º; T_ˇ DC^? DC^?N DR2˚R3DSpan¹2; 3º; TD®

XjX ?H; J X DJ1X¯

; TD®

XjX ?H; J X D J1X¯

;

whereR,C and H denote, respectively, the real, complex and quaternionic span of the structure vector field, andC^?denotes the orthogonal complement ofCinH.

Now let us check case by case whether both sides of () are equal to each other:

Case A-1.X 2T˛DSpan¹º.

By puttingXDin (), it can be easily checked that the two sides are equal.

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Case A-2.X 2T_ˇ DSpan¹2; 3º.

Then we put A2 D ˇ2, A3 D ˇ3, where ˇ D p 2cot.p

2r/. Then by puttingXD2in () we have

left sideDA12DA12DA121D A13DA2Dˇ2; right sideD1A2Dˇ12Dˇ121D ˇ13Dˇ2:

From this, both sides are equal toˇ2. Similarly, by puttingX D3in () we get that they are equal toˇ3.

Case A-3.Xi 2TD ¹X jX?H; X D1Xº,i D1; : : : ; 2.m 1/.

Then in () we assert

left sideDA1Xi DA₁²Xi D AXi D Xi; right sideD1AXi D1Xi D₁²Xi D Xi

foriD1; : : : ; 2.m 1/.

Case A-4.Xi 2TD ¹X jX?H; X D 1Xº,i D1; : : : ; 2.m 1/.

Then in () we have

left sideDA1Xi D A₁²Xi DAXi DXi D0;

right sideD1AXi D1Xi D0:

Summarizing these cases, we deduce the following result.

Remark 6.The shape operatorAfor real hypersurfaces of type (A) inG2.C^m^C²/ satisfies the commuting condition ().

Next we consider a tube of type (B) in G2.C^m^C²/, which is a tube around a totally geodesicHPⁿ,mD2n, inG2.C^m^C²/. That is, for a tube of type (B) in Theorem A we introduce the following proposition.

Proposition F([3]).LetMbe a connected real hypersurface ofG2.C^m^C²/. Sup- pose that AD D, A D ˛, and is tangent to D. Then the quaternionic dimensionmofG2.C^m^C²/is even, saymD2n, andMhas five distinct constant principal curvatures

˛D 2tan.2r/; ˇD2cot.2r/; D0; Dcot.r/; D tan.r/

with somer 2.0; =4/. The corresponding multiplicities are

m.˛/D1; m.ˇ/D3Dm. /; m./D4n 4Dm./

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and the corresponding eigenspaces are

T˛ DRDSpan¹º; Tˇ DJJ DSpan®

jD1; 2; 3¯

; T DJDSpan®

jD1; 2; 3¯

; T; T;

where

T˚TD.HC/^?; JTDT; JTDT; J TDT:

Now let us check for real hypersurfaces of type (B) whether they satisfy the property of commuting condition ().

Here we suppose that a real hypersurface of type (B) has the commuting shape operatorA, that is, the shape operatorAonM satisfies the commuting condition A1X D1AXfor any tangent vector fieldXonM. Then we putA1Dˇ1, whereˇ D2cot.2r/. And by puttingX D1in () we have

left sideDA11DA₁²D A D ˛;

right sideD1A1Dˇ11Dˇ₁²D ˇ:

From this, we obtain˛Dˇ. But this case can not occur for somer 2.0; =4/. In fact,˛ D 2tan.2r/ < 0andˇ D2cot.2r/ > 0. So we also give the following remark:

Remark 7.The shape operatorAof real hypersurfaces of type (B) inG2.C^m^C²/ does not satisfy the commuting condition ().

Therefore summing up Lemma 5, Theorem A and together with Remarks 6 and 7, we complete the proof of Theorem 1.

Next, we consider the case when the Reeb vector belongs to the distribution D^?. We assert the following:

Lemma 8.LetM be a real hypersurface inG2.C^m^C²/, m 3, satisfying the commuting shape operator (). If the Reeb vector belongs to the distribution D^?, then the distributionD^?is invariant under the shape operatorAofM, that is,g.AD;D^?/D0.

Proof. From now on, since 2 D^?, let us put D 1. Taking the covariant derivative along any directionX2TM, we have

AXD rXD rX1Dq3.X /2 q2.X /3C1AX: (3.3)

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From this, taking an inner product with2and3, we have

q3.X /D2g.AX; 3/; q2.X /D2g.AX; 2/; (3.4) respectively. By using (3.3) and (3.4), we have

AX D2g.AX; 3/2 2g.AX; 2/3C1AX

for any tangent vector fieldX onM. From this, applying the structure tensor, we obtain

AX D˛.X /C2g.AX; 2/2C2g.AX; 3/3 1AX; (3.5) where we have used the formulas (2.2) and (3.4).

PuttingX D2in (3.5), we get

A2D2g.A2; 2/2C2g.A2; 3/3 1A2: (3.6) By taking an inner product with anyX 2Din (3.6) and using the assumption (), we have

g.A2; X /D g.1A2; X /D g.A1X; 2/ D g.1AX; 2/D g.AX; 2/:

Then for anyX2D, we get

g.AX; 2/D0: (3.7) Similarly, if we substitute forXD3in (3.5), we get

A3D2g.A3; 2/2C2g.A3; 3/3 1A3: (3.8) From this, by taking an inner product with any X 2 Din (3.8) and using the assumption (), we have

g.AX; 3/D0 (3.9) for any X 2 D. From the above two equations (3.7), (3.9) and the result of Lemma 5, we have g.AX; / D 0 for D 1; 2; 3 and anyX 2 D, that is, g.AD;D^?/D0. This gives a complete proof of the lemma.

Therefore summing up Lemmas 5 and 8, Theorem A and together with Re- marks 6 and 7 mentioned above, we complete the proof of Theorem 2.

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4 Proof of Theorems 3 and 4

Throughout this section, we assume thatM is a connected real hypersurfaceM in G2.C^m^C²/,m3, with the commuting shape operator, that is, the shape operator satisfies the condition ().

Now we consider the case that the principal curvature˛of the Reeb vector field is constant. Then we obtain the following lemma.

Lemma 9.LetM be a connected real hypersurface inG2.C^m^C²/,m 3, satisfying commuting condition(). If the principal curvature˛of the Reeb vector fieldis constant, then the Reeb vector fieldbelongs to either the distributionD or the distributionD^?.

Proof. From the assumption () and the result of Lemma 5, we know thatA D

˛, where˛ D .A/. Then by differentiatingA D ˛ and using the equation of Codazzi in Section 2, we have

Y ˛D. ˛/.Y / 4

3

X

D1

./.Y /

for any Y 2 TM (see [4]). From this, by applying the assumption that ˛ is constant, we have

3

X

D1

./.Y /D0 (4.1)

for anyY 2TM. Let us putD.X0/X0C.1/1for some unitX02Dand non-vanishing functions.X0/and.1/.

For anyY 2TM, equation (4.1) yields 1./1.Y /D0:

Since.1/is not zero, we have1D0. Then we obtain1X0D0. This gives a contradiction, thus completing the proof of Lemma 9.

In Lemma 9 we may consider the two cases that 2 Dand 2 D^?, respectively.

First, let us consider the case that2D^?. By Lemmas 5 and 8, we see thatM is locally congruent to a real hypersurface of type (A) under our assumption. And in Section 3 we have checked that the shape operatorAof a real hypersurface of type (A) satisfies the condition () (see Remark 6).

Next we consider the case 2 D. For the case that the Reeb vector field belongs to the distributionD, we introduce a lemma due to Lee and Suh [5] as follows:

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Lemma 10([5]).LetM be a connected Hopf hypersurface inG2.C^m^C²/,m3.

If2D, theng.AD;D^?/D0.

So by Lemmas 5 and 10, we see thatM is locally congruent to a real hypersurface of type (B) under our assumption. But in Section 3 we have checked that the shape operatorAof a real hypersurface of type (B) does not satisfy the condition () (see Remark 7). From these facts, we complete the proof of Theorem 3.

On the other hand, we consider the normal Jacobi operatorRNN. In [6], Pérez, Jeong and Suh have classified real hypersurfaces inG2.C^m^C²/with two commuting conditions with respect to the normal Jacobi operator, that is,RNNı Dı NRN

orRNN ıADAı NRN.

Now by putting a unit normal vector fieldN into the curvature tensorRN of the ambient spaceG2.C^m^C²/, we calculate the normal Jacobi operatorRNN in such a way that

RNNX D NR.X; N /N

DXC3.X /C3

3

X

D1

.X / 3

X

D1

®./.X .X // .X /¯

for any tangent vector fieldX onM inG2.C^m^C²/. That normal Jacobi operator RNN commuteswith the shape operatorA(or the structure tensor) ofM in G2.C^m^C²/means that the eigenspace of the normal Jacobi operator isinvariant by the shape operatorA(or the structure tensor). So we deduce the following lemma.

Lemma 11.IfMis a real hypersurface inG2.C^m^C²/,m3, satisfyingRNNı D ı NRN, then2Dor 2D^?.

Hence, together with Lemmas 5, 8 and 10, and Remarks 6 and 7, we have a complete proof of Theorem 4.

Acknowledgments. The authors would like to express their sincere gratitude to the referee for carefully reading our manuscript and useful suggestions to improve the first version of this paper.

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Received March 15, 2012; revised May 11, 2012.

Author information

Imsoon Jeong, Department of Mathematics, Kyungpook National University, Taegu 702-701, Republic of Korea.

E-mail:[email protected]

Young Jin Suh, Department of Mathematics, Kyungpook National University, Taegu 702-701, Republic of Korea.

E-mail:[email protected]