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Real submanifolds of codimension two of a complex space form

Mirjana Djori´ c, Masafumi Okumura PADGE 2012 workshop August 28, 2012. Leuven, Belgium

Mirjana Djori´ c, Masafumi Okumura Real submanifolds of codimension two of a complex space form

The study of real hypersurfaces of K¨ ahlerian manifolds has been an important subject in geometry of submanifolds, especially when the ambient space is a complex space form.

Mirjana Djori´ c, Masafumi Okumura Real submanifolds of codimension two of a complex space form

The study of real hypersurfaces of K¨ ahlerian manifolds has been an important subject in geometry of submanifolds, especially when the ambient space is a complex space form.

• R. Niebergall and P.J. Ryan,

Real hypersurfaces in complex space forms, in Tight and taut submanifolds, (eds. T.E. Cecil and S. S. Chern), Math.Sciences Res. Inst. Publ. 32, Cambridge Univ. Press, Cambridge, 233–305, (1997).

Mirjana Djori´ c, Masafumi Okumura Real submanifolds of codimension two of a complex space form

However, for arbitrary codimension, there are only a few recent results.

Mirjana Djori´ c, Masafumi Okumura Real submanifolds of codimension two of a complex space form

However, for arbitrary codimension, there are only a few recent results.

• M. Djori´

c, M. Okumura, CR submanifolds of complex projective space,

• Develop. in Math. 19, Springer, (2009).

Mirjana Djori´ c, Masafumi Okumura Real submanifolds of codimension two of a complex space form

1฀3

developments in mathematics

19

19

devm

Mirjana Djorić · Masafumi Okumura CR Submanifolds of Complex Projective Space

CR Submanifolds

• f Complex

Projective Space

CR Submanifolds of Complex Projective Space Djorić · Okumura

1

This book covers the necessary topics for learning the basic properties of complex manifolds and their submanifolds, offering an easy, friendly, and accessible introduction into the subject while aptly guiding the reader to topics of current research and to more advanced publications. The book begins with an introduction to the geometry of complex manifolds and their submanifolds and describes the properties of hypersurfaces and CR submanifolds, with particular emphasis on CR submanifolds of maximal CR dimension. The second part contains results which are not new, but recently published in some mathematical journals. The final part contains several original results by the authors, with complete proofs. Key features of CR Submanifolds of Complex Projective Space:

• Presents recent developments and results in the study of submanifolds previously

published only in research papers.

• Special topics explored include: the Kähler manifold, submersion and immersion,

codimension reduction of a submanifold, tubes over submanifolds, geometry

• f hypersurfaces and CR submanifolds of maximal CR dimension.
• Provides relevant techniques, results and their applications, and presents insight

into the motivations and ideas behind the theory.

• Presents the fundamental definitions and results necessary for reaching the frontiers
• f research in this field.

This text is largely self-contained. Prerequisites include basic knowledge of introductory manifold theory and of curvature properties of Riemannian geometry. Advanced undergraduates, graduate students and researchers in differential geometry will benefit from this concise approach to an important topic.

Mirjana Djorić Masafumi Okumura

Developments in Mathematics is a book series devoted to all areas of mathematics, pure and applied. The series emphasizes research monographs describing the latest

• advances. Edited volumes that focus on areas that have seen dramatic progress, or are
• f special interest, are encouraged as well.

ISBN 978-1-4419-0433-1 9 7 8 1 4 4 1 9 0 4 3 3 1

Mirjana Djori´ c, Masafumi Okumura Real submanifolds of codimension two of a complex space form

Let M be an n–dimensional real submanifold of

Mirjana Djori´ c, Masafumi Okumura Real submanifolds of codimension two of a complex space form

Let M be an n–dimensional real submanifold of an almost Hermitian manifold M

n+p

Mirjana Djori´ c, Masafumi Okumura Real submanifolds of codimension two of a complex space form

Let M be an n–dimensional real submanifold of an almost Hermitian manifold M

n+p

with the immersion ı : M → M

Mirjana Djori´ c, Masafumi Okumura Real submanifolds of codimension two of a complex space form

Let M be an n–dimensional real submanifold of an almost Hermitian manifold M

n+p

with the immersion ı : M → M g(X, Y ) = g(ıX, ıY ), X, Y ∈ T(M)

Mirjana Djori´ c, Masafumi Okumura Real submanifolds of codimension two of a complex space form

Hx(M) = Tx(M) ∩ JTx(M) is called the holomorphic tangent space of M.

Mirjana Djori´ c, Masafumi Okumura Real submanifolds of codimension two of a complex space form

Hx(M) = Tx(M) ∩ JTx(M) is called the holomorphic tangent space of M. Hx(M) is the maximal J-invariant subspace of Tx(M).

Mirjana Djori´ c, Masafumi Okumura Real submanifolds of codimension two of a complex space form

Hx(M) = Tx(M) ∩ JTx(M) is called the holomorphic tangent space of M. Hx(M) is the maximal J-invariant subspace of Tx(M). n − p ≤ dimRHx(M) ≤ n

Mirjana Djori´ c, Masafumi Okumura Real submanifolds of codimension two of a complex space form

M is called the Cauchy-Riemann submanifold

• r briefly CR submanifold

Mirjana Djori´ c, Masafumi Okumura Real submanifolds of codimension two of a complex space form

M is called the Cauchy-Riemann submanifold

• r briefly CR submanifold if Hx has constant dimension for any

x ∈ M.

• R. Nirenberg and R.O. Wells, Jr., Approximation theorems
• n differentiable submanifolds of a complex manifold, Trans.
• Amer. Math. Soc. 142, 15–35, (1965).

Mirjana Djori´ c, Masafumi Okumura Real submanifolds of codimension two of a complex space form

Examples (CR submanifolds of a complex manifold)

Mirjana Djori´ c, Masafumi Okumura Real submanifolds of codimension two of a complex space form

Examples (CR submanifolds of a complex manifold) J-invariant submanifolds.

Mirjana Djori´ c, Masafumi Okumura Real submanifolds of codimension two of a complex space form

Examples (CR submanifolds of a complex manifold) J-invariant submanifolds. JıTx(M) ⊂ ıTx(M),

Mirjana Djori´ c, Masafumi Okumura Real submanifolds of codimension two of a complex space form

Examples (CR submanifolds of a complex manifold) J-invariant submanifolds. JıTx(M) ⊂ ıTx(M), Hx(M) = Tx(M) , dimRHx(M) = n . Real hypersurfaces.

Mirjana Djori´ c, Masafumi Okumura Real submanifolds of codimension two of a complex space form

Examples (CR submanifolds of a complex manifold) J-invariant submanifolds. JıTx(M) ⊂ ıTx(M), Hx(M) = Tx(M) , dimRHx(M) = n . Real hypersurfaces. dimRHx(M) = n − 1.

Mirjana Djori´ c, Masafumi Okumura Real submanifolds of codimension two of a complex space form

Examples (CR submanifolds of a complex manifold) J-invariant submanifolds. JıTx(M) ⊂ ıTx(M), Hx(M) = Tx(M) , dimRHx(M) = n . Real hypersurfaces. dimRHx(M) = n − 1. Totally real submanifolds.

Mirjana Djori´ c, Masafumi Okumura Real submanifolds of codimension two of a complex space form

Examples (CR submanifolds of a complex manifold) J-invariant submanifolds. JıTx(M) ⊂ ıTx(M), Hx(M) = Tx(M) , dimRHx(M) = n . Real hypersurfaces. dimRHx(M) = n − 1. Totally real submanifolds. Hx(M) = {0} holds at every point x ∈ M.

Mirjana Djori´ c, Masafumi Okumura Real submanifolds of codimension two of a complex space form

Let Mn be a CR submanifold of maximal CR dimension dimR(JTx(M) ∩ Tx(M)) = n − 1 at each point x of M

Mirjana Djori´ c, Masafumi Okumura Real submanifolds of codimension two of a complex space form

Let Mn be a CR submanifold of maximal CR dimension dimR(JTx(M) ∩ Tx(M)) = n − 1 at each point x of M Then it follows that M is odd–dimensional and that there exists a unit vector field ξ normal to M such that JTx(M) ⊂ Tx(M) ⊕ span{ξx} for any x ∈ M

Mirjana Djori´ c, Masafumi Okumura Real submanifolds of codimension two of a complex space form

Examples

real hypersurfaces of almost Hermitian manifolds M;

Mirjana Djori´ c, Masafumi Okumura Real submanifolds of codimension two of a complex space form

Examples

real hypersurfaces of almost Hermitian manifolds M; real hypersurfaces M of complex submanifolds M′ of almost Hermitian manifolds M;

Mirjana Djori´ c, Masafumi Okumura Real submanifolds of codimension two of a complex space form

Examples

real hypersurfaces of almost Hermitian manifolds M; real hypersurfaces M of complex submanifolds M′ of almost Hermitian manifolds M;

• dd-dimensional F ′-invariant submanifolds M of real

hypersurfaces M′ of almost Hermitian manifolds M,

Mirjana Djori´ c, Masafumi Okumura Real submanifolds of codimension two of a complex space form

Examples

real hypersurfaces of almost Hermitian manifolds M; real hypersurfaces M of complex submanifolds M′ of almost Hermitian manifolds M;

• dd-dimensional F ′-invariant submanifolds M of real

hypersurfaces M′ of almost Hermitian manifolds M, where F ′ is an almost contact metric structure naturally induced by the almost Hermitian structure on M.

Mirjana Djori´ c, Masafumi Okumura Real submanifolds of codimension two of a complex space form

Submanifolds of codimension 2 of a complex manifold

Let M be a real (n + 2)-dimensional complex manifold, J its natural almost complex structure and g its Hermitian metric.

Mirjana Djori´ c, Masafumi Okumura Real submanifolds of codimension two of a complex space form

Submanifolds of codimension 2 of a complex manifold

Let M be a real (n + 2)-dimensional complex manifold, J its natural almost complex structure and g its Hermitian metric. We consider an n-dimensional submanifold M of M and we denote by ı the immersion ı : M → M.

Mirjana Djori´ c, Masafumi Okumura Real submanifolds of codimension two of a complex space form

Submanifolds of codimension 2 of a complex manifold

Let M be a real (n + 2)-dimensional complex manifold, J its natural almost complex structure and g its Hermitian metric. We consider an n-dimensional submanifold M of M and we denote by ı the immersion ı : M → M. An n-dimensional complex hypersurface, which is a CR submanifold of CR dimension n−2

2 , is a real submanifold of

codimension two, but there also exist real submanifolds of codimension two which are not CR submanifolds.

Mirjana Djori´ c, Masafumi Okumura Real submanifolds of codimension two of a complex space form

We consider an n-dimensional submanifold M of M and we denote by ı the immersion ı : M → M. We choose mutually orthogonal unit normals to M and we denote them by ξ1 and ξ2 . Then JıX = ıFX + u1(X)ξ1 + u2(X)ξ2, Jξ1 = −ıU1 + λξ2, Jξ2 = −ıU2 − λξ1.

Mirjana Djori´ c, Masafumi Okumura Real submanifolds of codimension two of a complex space form

We note that u1 and u2 depend on the choice of normals ξ1 and ξ2, but the function λ2 = g(Jξ1, ξ2) does not depend on the choice

• f ξ1 and ξ2.

Mirjana Djori´ c, Masafumi Okumura Real submanifolds of codimension two of a complex space form

Since (M, J, g) is a Hermitian manifold, we compute F 2X = −X + u1(X)U1 + u2(X)U2, FU1 = −λU2, FU2 = λU1, g(Ua, X) = ua(X), a = 1, 2, g(U1, U1) = g(U2, U2) = 1 − λ2, g(U1, U2) = 0.

Mirjana Djori´ c, Masafumi Okumura Real submanifolds of codimension two of a complex space form

Proposition

For the function λ, we have the following: (1) M is a complex hypersurface if and only if λ2(x) = 1 for any x ∈ M. (2) M is a CR submanifold of CR dimension n−2

2

if λ(x) = 0 for any x ∈ M.

Mirjana Djori´ c, Masafumi Okumura Real submanifolds of codimension two of a complex space form

• Remark. In (2) of the previous Proposition, the converse is not

true, that is, a CR submanifold of CR dimension n−2

2

does not always satisfy λ = 0.

Mirjana Djori´ c, Masafumi Okumura Real submanifolds of codimension two of a complex space form

Example.

Let the ambient manifold be a complex Euclidean space Cm+1 and Mn, n = 2m be a submanifold defined by Rezm+1 = Imzm, Imzm+1 = 0, that is, using real coordinate system (x1, y1, . . . , xm+1, ym+1), M is defined by (x1, y1, . . . , xm−1, ym−1, xm, ym, ym, 0). M is a CR submanifold of CR dimension n−2

2

and the mutually

• rthonormal normal vectors to M are

ξ1 = ∂ ∂ym+1 , ξ2 = 1 √ 2 ( ∂ ∂ym − ∂ ∂xm+1 ). Thus, λ =< Jξ1, ξ2 >=

1 √ 2.

Mirjana Djori´ c, Masafumi Okumura Real submanifolds of codimension two of a complex space form

The Gauss and Weingarten formulae are the following. ∇XıY = ı∇XY + h(X, Y ) = ı∇XY + g(A1X, Y )ξ1 + g(A2X, Y )ξ2 ∇Xξ1 = −ıA1X + s(X)ξ2, ∇Xξ2 = −ıA2X − s(X)ξ1, where h(X, Y ) is the second fundamental form, Aa the shape

• perator with respect to the normal ξa, a = 1, 2 and s the third

fundamental form.

Mirjana Djori´ c, Masafumi Okumura Real submanifolds of codimension two of a complex space form

We assume that the ambient manifold M is a K¨ ahler manifold. Then, since ∇J = 0, it follows (∇XF)Y =

2

• a=1

{ua(Y )AaX − g(AaX, Y )Ua}, ∇XU1 = FA1X − λA2X + s(X)U2, ∇XU2 = FA2X + λA1X − s(X)U1, Xλ = g(A2U1 − A1U2, X).

Mirjana Djori´ c, Masafumi Okumura Real submanifolds of codimension two of a complex space form

Now we assume that M satisfies the condition h(FX, Y ) + h(X, FY ) = 0. Then, it is easily seen that this condition is equivalent to AaF = FAa a = 1, 2, that is, the linear map F commutes with the both shape operators A1 and A2.

Mirjana Djori´ c, Masafumi Okumura Real submanifolds of codimension two of a complex space form

Theorem If a complex hypersurface M of a K¨ ahler manifold M satisfies the condition h(FX, Y ) + h(X, FY ) = 0, then M is a totally geodesic submanifold.

Mirjana Djori´ c, Masafumi Okumura Real submanifolds of codimension two of a complex space form

We consider the following open submanifold of M defined by M0 = {x ∈ M|λ(x)(λ(x)2 − 1) = 0}.

Mirjana Djori´ c, Masafumi Okumura Real submanifolds of codimension two of a complex space form

We consider the following open submanifold of M defined by M0 = {x ∈ M|λ(x)(λ(x)2 − 1) = 0}. Lemma If the condition h(FX, Y ) + h(X, FY ) = 0, is satisfied, then in M0, U1 and U2 are eigenvectors of both A1 and A2.

Mirjana Djori´ c, Masafumi Okumura Real submanifolds of codimension two of a complex space form

We consider the following open submanifold of M defined by M0 = {x ∈ M|λ(x)(λ(x)2 − 1) = 0}. Lemma If the condition h(FX, Y ) + h(X, FY ) = 0, is satisfied, then in M0, U1 and U2 are eigenvectors of both A1 and A2. AaU1 = αaU1, AaU2 = αaU2, a = 1, 2.

Mirjana Djori´ c, Masafumi Okumura Real submanifolds of codimension two of a complex space form

Submanifolds of a complex space form

From now on, we assume that the ambient manifold M is a complex space form, i.e. a complex manifold of constant holomorphic sectional curvature 4k. Then the Codazzi equation becomes (∇XAa)Y − (∇Y Aa)X = k{ua(X)FY − ua(Y )FX − 2g(FX, Y )Ua} +

2

• b=1

{sab(X)AbY − sab(Y )AbX}, where s = s12 = −s21.

Mirjana Djori´ c, Masafumi Okumura Real submanifolds of codimension two of a complex space form

Lemma In M0, the eigenvalues α1 and α2 satisfy the following equations. Xα1 − α2s(X) = −3kλu2(X), Xα2 + α1s(X) = 3kλu1(X)

Mirjana Djori´ c, Masafumi Okumura Real submanifolds of codimension two of a complex space form

Lemma Under the above conditions, if the complex space form M is not a complex Euclidean space, then M0 = ∅.

Mirjana Djori´ c, Masafumi Okumura Real submanifolds of codimension two of a complex space form

Theorem Let M be a non Euclidean complex space form. If a real submanifold M of codimension 2 satisfies the condition h(FX, Y ) + h(X, FY ) = 0, then one of the following holds. (1) M is a totally geodesic complex hypersurface. (2) M is a CR submanifold of CR dimension n−2

2

with λ = 0, where λ is a function defined on the submanifold M.

Mirjana Djori´ c, Masafumi Okumura Real submanifolds of codimension two of a complex space form

Submanifolds of a complex Euclidean space

Now we consider the open submanifold M0 of codimension 2 of a complex Euclidean space. Lemma In M0, α2

1 + α2 2 is constant.

Mirjana Djori´ c, Masafumi Okumura Real submanifolds of codimension two of a complex space form

Now we consider the case that α2

1 + α2 2 = 0.

Mirjana Djori´ c, Masafumi Okumura Real submanifolds of codimension two of a complex space form

Now we consider the case that α2

1 + α2 2 = 0.

In this case, we put ξ′

1

= 1

• α2

1 + α2 2

(α1ξ1 + α2ξ2), ξ′

2

= 1

• α2

1 + α2 2

(α2ξ1 − α1ξ2) and compute A′

1X

= 1

• α2

1 + α2 2

(α1A1 + α2A2)X, A′

2X

= 1

• α2

1 + α2 2

(α2A1 − α1A2)X, s′(X) = 0,

Mirjana Djori´ c, Masafumi Okumura Real submanifolds of codimension two of a complex space form

Now we consider the case that α2

1 + α2 2 = 0.

In this case, we put ξ′

1

= 1

• α2

1 + α2 2

(α1ξ1 + α2ξ2), ξ′

2

= 1

• α2

1 + α2 2

(α2ξ1 − α1ξ2) and compute A′

1X

= 1

• α2

1 + α2 2

(α1A1 + α2A2)X, A′

2X

= 1

• α2

1 + α2 2

(α2A1 − α1A2)X, s′(X) = 0, which means that we have chosen the orthonormal normals ξ′

1 and

ξ′

2 in such a way that the normal connection is trivial.

Mirjana Djori´ c, Masafumi Okumura Real submanifolds of codimension two of a complex space form

From the relations above we compute that the corresponding eigenvalues α′

1, α′ 2 of A′ a for U′ a are

α′

1 =

• α2

1 + α2 2,

α′

2 = 0.

Since in the previous discussion we chose a set of arbitrary

• rthonormal normals ξ1 and ξ2, the corresponding relations must

be satisfied for the new chosen orthonormal normals ξ′

1 and ξ′ 2:

α′

1A′ 2X = α′ 2A′ 1X.

Hence A′

2X = 0.

Mirjana Djori´ c, Masafumi Okumura Real submanifolds of codimension two of a complex space form

Lemma If α2

1 + α2 2 = 0, then there exists an (n + 1)-dimensional totally

geodesic Euclidean subspace En+1 of C

n+2 2

such that M0 is a hypersurface of En+1.

Mirjana Djori´ c, Masafumi Okumura Real submanifolds of codimension two of a complex space form

Lemma If α2

1 + α2 2 = 0, then there exists an (n + 1)-dimensional totally

geodesic Euclidean subspace En+1 of C

n+2 2

such that M0 is a hypersurface of En+1. According to this Lemma, we can regard the submanifold M0 as a hypersurface of En+1.

Mirjana Djori´ c, Masafumi Okumura Real submanifolds of codimension two of a complex space form

Theorem Let M be a connected real submanifold of codimension 2 of a complex Euclidean space C

n+2 2

which satisfies the condition h(FX, Y ) + h(X, FY ) = 0. Then M is one of the following: (1) n-dimensional Euclidean space En; (2) n-dimensional sphere Sn; (3) product manifold of an r-dimensional sphere with an (n − r)-dimensional Euclidean space Sr × En−r, where r is an even number; (4) M is a CR submanifold of CR dimension n−2

2

with λ = 0.

Mirjana Djori´ c, Masafumi Okumura Real submanifolds of codimension two of a complex space form

Theorem Let Mn be a real submanifold of codimension two of a complex Euclidean space C

n+2 2

with λ = 0 which satisfies the condition h(FX, Y ) + h(X, FY ) = 0. hypersurface M′ of C

n+2 2 , i.e. A′ = cI, c = 0, such that M ⊂ M′,

then M is a product of two odd-dimensional spheres.

Mirjana Djori´ c, Masafumi Okumura Real submanifolds of codimension two of a complex space form

Theorem Let M be a real submanifold of codimension two of a complex Euclidean space C

n+2 2

with λ = 0 which satisfies the condition h(FX, Y ) + h(X, FY ) = 0. If there exists a totally geodesic hypersurface M′ of C

n+2 2

such that M ⊂ M′, then M is one of the following: (1) n-dimensional hyperplane En, (2) product manifold of an odd-dimensional sphere and a Euclidean space: S2p+1 × En−2p−1.

Mirjana Djori´ c, Masafumi Okumura Real submanifolds of codimension two of a complex space form

To appear in

Mirjana Djori´ c, Masafumi Okumura Real submanifolds of codimension two of a complex space form

To appear in Differential Geometry and its Applications

Mirjana Djori´ c, Masafumi Okumura Real submanifolds of codimension two of a complex space form