It is proved here that a minimal isometric immersion of a Kähler-Einstein or homogeneous Kähler-manifold into an Euclidean space must be totally geodesic

Di Scala, Antonio Jose'

English

It is proved here that a minimal isometric immersion of a Kähler-Einstein or homogeneous Kähler-manifold into an Euclidean space must be totally geodesic

DI SCALA, ANTONIO JOSE'

PORTO@iris (Publications Open Repository TOrino - Politecnico di Torino)

09 November 2022POLITECNICO DI TORINORepository ISTITUZIONALEMinimal Immersions of Kahler manifolds into Euclidean Spaces / DI SCALA, ANTONIO JOSE'. - In: BULLETIN OF THELONDON MATHEMATICAL SOCIETY. - ISSN 0024-6093. - STAMPA. - 35:(2003), pp. 825-827.[10.1112/S0024609303002492]OriginalMinimal Immersions of Kahler manifolds into Euclidean SpacesPublisher:PublishedDOI:10.1112/S0024609303002492Terms of use:openAccessPublisher copyright(Article begins on next page)This article is made available under terms and conditions as specified in the  corresponding bibliographic description inthe repositoryAvailability:This version is available at: 11583/1399743 since: 2015-11-30T15:07:01ZOxford University PressSubmitted exclusively to the London Mathematical SocietyDOI: 10.1112/S0000000000000000MINIMAL IMMERSIONS OF KÄHLER MANIFOLDS INTOEUCLIDEAN SPACESANTONIO J. DI SCALAPost print (i.e. final draft post-refereeing) version of an article published on Bull. London Math. Soc. 35 (2003), no. 6, 825827.doi: 10.1112/S0024609303002492 .Beyond the journal formatting, please note that there could be minor changes from this document to the final published version.The final published version is accessible from here:http://blms.oxfordjournals.org/content/35/6/825.full.pdf+htmlThis document has made accessible through PORTO, the Open Access Repository of Politecnico di Torino (http://porto.polito.it),in compliance with the Publisher’s copyright policy as reported in the SHERPA-ROMEO website:http://www.sherpa.ac.uk/romeo/search.php?issn=0024-6093AbstractWe prove that a minimal isometric immersion of a Kähler-Einstein or homogeneous Kähler man-ifold into an Euclidean space must be totally geodesic. As an application we show that an opensubset of the real hyperbolic plane RH2 cannot be minimally immersed into the Euclidean space.As another application we prove that if an irreducible Kähler manifold is minimally immersed inan Euclidean space then its restricted holonomy group must be U(n), where n = dimCM .1. IntroductionThe aim of this article is to prove the following results.Theorem 1. Let M be a Kähler manifold (connected, non necessarily complete)that is either homogeneous or Einstein. Then, any minimal isometric immersionf : M → Rn is totally geodesic, i.e. f(M) is an open subset of an affine subspaceof Rn.Recall that a Riemannian manifold with parallel Ricci curvature tensor is locallya product of Einstein manifolds. Since minimal immersions of Riemannian productsinto a Euclidean space are product immersions [9] we obtain the following corollary.Theorem 2. Let M be a Kähler manifold that is a product M = H × E ofan homogeneous Kähler manifold H and a Kähler manifold E with parallel Riccitensor. Then, any minimal isometric immersion f : M → Rn is totally geodesic.Since any orientable surface is naturally a Kähler manifold we obtain the followingwell-known corollary [2, Proposition 4.1], [10, Chapter IV].Corollary 1. Let U ⊂ RH2 be an open submanifold of the real hyperbolicplane RH2. Then, there is no minimal isometric immersion f : U → Rn.Finally, we obtain the following corollary which is a generalization of the re-sult in [12] concerning the restricted holonomy group of hypersurfaces in complexEuclidean spaces.2000 Mathematics Subject Classification 53B25 (primary), 53C42 (secondary).Supported by an EPSRC Grant GR/R69174.2 antonio j. di scalaCorollary 2. Let Mn be a locally irreducible Kähler manifold of complex di-mension n and assume that there exists a minimal isometric immersion f : M →R2n+m. Then, the restricted holonomy group of M is U(n). In particular, the re-stricted holonomy group of an irreducible complex submanifold of Cn+m is U(n).It interesting to note that a strong version of the last statement of the abovecorollary also holds for Kähler manifolds Mn isometrically and holomorphicallyimmersed in the complex hyperbolic space CHn+p. Namely, a Kähler manifold Mnisometrically and holomorphically immersed in CHn+p is locally irreducible and itsholonomy group Hol(∇) is U(n) (see [1]).2. Proof of the main theorem and further remarksTo prove our main theorem we note that Theorem 1.2 from [6] and Theorem 1.11from [5] implies the following theorem.Theorem 3. [6] [5] Let M be a simple connected Kähler manifold (not neces-sary complete) and let f : M → Rn be a minimal isometric immersion. Then thereexist a minimal isometric immersion g : M → Rn such that f : M → Cn = Rn×Rnis given by f(p) = ( f(p)√2, g(p)√2) is isometric and holomorphic with respect to the com-plex structure J(u, v) = (−v, u) on Rn × Rn.Recall the following remarkable result by M. Umehara.Theorem 4. [13] Every Kähler-Einstein submanifold of a complex linear orhyperbolic space is always totally geodesic.Notice that the above theorem is of local nature. The proof is by using the socalled “diastatic function” introduced by E. Calabi in [4].We need the following result.Theorem 5. [7] Let M be an homogeneous Kähler manifold and f : M → Cnan holomorphic and isometric immersion. Then f is totally geodesic.Now we can prove our main theorem.Proof of Theorem 1. Let M̃ be the universal covering of M and f̃ = f ◦π, whereπ : M̃ → M is the canonical projection. Let f̃ : M̃ → Cn be the isometric andholomorphic immersion given by Theorem 3. Notice that f is totally geodesic if andonly if f̃ is totally geodesic.Assume now that M is Kähler-Einstein. Then M̃ is Kähler-Einstein. If p ∈ M̃ isany point of M̃ then we can restrict f̃ to an small neighborhood U of p ∈ M̃ suchthat f̃ |U : U → Cn is an imbedding. Then Theorem 4 implies that f̃ |U is totallygeodesic. So f̃ is totally geodesic.Assume now that M is homogeneous. Then M̃ is homogeneous. Then by Theoremminimal immersions of kähler manifolds 35 it follows that f̃ is totally geodesic. This completes the proof of our main theorem.Proof of Corollary 2. Observe that f can not be totally geodesic because weassume that M is locally irreducible. Theorem 2 shows that the Ricci tensor of Mcan not be parallel. Let Φ∗ be the restricted holonomy group of M . The classifi-cation of irreducible restricted holonomy groups by M. Berger shows that eitherΦ∗ is U(n), SU(n), Sp(n) or M is a locally symmetric space [3]. If M is a locallysymmetric space we get that M has parallel Ricci tensor. If Φ∗ is one of the groupsSU(n), Sp(n) we get that M is Ricci-flat. Thus Φ∗ = U(n), as we have to prove.Let M be a Riemannian manifold and let f : M → Rn be a minimal isometricimmersion. It is interesting to note that f must be totally geodesic if one of thefollowing conditions holds.(1) M is complete, the Ricci tensor of M has constant eigenvalues and f has flatnormal bundle (see [8]),(2) M is Ricci-flat (as follows from Gauss equation),(3) f has codimension 2 and M is Einstein (see [11]).So, it seems natural to pose the following conjecture.Conjecture. Let M be a Riemannian manifold that is either locally homoge-neous or Einstein. Then, any minimal isometric immersion f : M → Rn must betotally geodesic.Acknowledgements. I wish to thank Marcos Dajczer for the helpful discussionsand comments. I also wish to thank Dmitri Alekseevsky and Kevin Baron for readthe manuscript and help me in improving it.References1. D.V. Alekseevsky and A. J. Di Scala, ‘The normal holonomy group of Kähler submani-folds’, Preprint (2002).2. R. Bryant, ‘Minimal surfaces of constant curvature in Sn’, Trans. A.M.S. 290 (1985) 1259–271.3. A.L. Besse, ‘Einstein Manifolds’, Ergebnisse der Mathematik und ihrer Grenzgebiete 3.Folge - Band 10, Springer Verlag (1987).4. E. Calabi, ‘Isometric imbedding of complex manifolds’, Ann. of Math. 58 (1953) 1–23.5. M. Dajczer and D. Gromoll, ‘Real Kaehler submanifolds and uniqueness of the Gaussmap’, J. Differential Geom. 22 (1985) 1 13–28.6. M. Dajczer and L. Rodriguez, ‘Rigidity of real Kaehler submanifolds’, Duke Math. J. 53(1986) 1 211–220.7. A.J. Di Scala, ‘Minimal homogeneous submanifolds in Euclidean spaces’, Ann. Global Anal.Geom. 21 (2002) 1 15–18.8. A.J. Di Scala and C. Olmos, ‘Submanifolds with curvature normals of constant length andthe Gauss map’, Preprint (2002).9. N. Ejiri, ‘Minimal immersions of Riemannian products into real space forms’, Tokyo J. Math.Vol. 2 Nro. 1 (1979) 63–70.10. H.B. Lawson, ‘Lectures on minimal submanifolds. Vol. I’, Mathematics Lecture Series, 9.Berkeley, California: Publish Perish Inc. (1980).11. Y. Matsuyama, ‘Minimal Einstein submanifolds with codimension two’, Tensor (N.S.) 52(1993) no. 1 61–68.4 minimal immersions of kähler manifolds12. K. Nomizu and B. Smyth, ‘Differential geometry of complex hypersurfaces II’, J. Math. Soc.Japan 20 (1968) 498–521.13. M. Umehara, ‘Einstein Kähler submanifolds of a complex linear or hyperbolic space’, TôhokuMath. Journ. 39 (1987) 385–389.Antonio J. Di ScalaThe University of HullCottingham RoadHU6 7RX, Hull, UKA.J.Di-Scala@maths.hull.ac.uk

Minimal Immersions of Kahler manifolds into Euclidean Spaces

Antonio J. Di Scala

Crossref

MINIMAL IMMERSIONS OF KÄHLER MANIFOLDS INTO EUCLIDEAN SPACES

PORTO Publications Open Repository TOrino

Submitted exclusively to the London Mathematical Society
DOI: 10.1112/S0000000000000000
MINIMAL IMMERSIONS OF KA¨HLER MANIFOLDS INTO
EUCLIDEAN SPACES
ANTONIO J. DI SCALA
Post print (i.e. final draft post-refereeing) version of an article published on Bull. London Math. Soc. 35 (2003), no. 6, 825827.
doi: 10.1112/S0024609303002492 .
Beyond the journal formatting, please note that there could be minor changes from this document to the final published version.
The final published version is accessible from here:
http://blms.oxfordjournals.org/content/35/6/825.full.pdf+html
This document has made accessible through PORTO, the Open Access Repository of Politecnico di Torino (http://porto.polito.it),
in compliance with the Publisher’s copyright policy as reported in the SHERPA-ROMEO website:
http://www.sherpa.ac.uk/romeo/search.php?issn=0024-6093
Abstract
We prove that a minimal isometric immersion of a Ka¨hler-Einstein or homogeneous Ka¨hler man-
ifold into an Euclidean space must be totally geodesic. As an application we show that an open
subset of the real hyperbolic plane RH2 cannot be minimally immersed into the Euclidean space.
As another application we prove that if an irreducible Ka¨hler manifold is minimally immersed in
an Euclidean space then its restricted holonomy group must be U(n), where n = dimCM .
1. Introduction
The aim of this article is to prove the following results.
Theorem 1. Let M be a Ka¨hler manifold (connected, non necessarily complete)
that is either homogeneous or Einstein. Then, any minimal isometric immersion
f : M → Rn is totally geodesic, i.e. f(M) is an open subset of an affine subspace
of Rn.
Recall that a Riemannian manifold with parallel Ricci curvature tensor is locally
a product of Einstein manifolds. Since minimal immersions of Riemannian products
into a Euclidean space are product immersions [9] we obtain the following corollary.
Theorem 2. Let M be a Ka¨hler manifold that is a product M = H × E of
an homogeneous Ka¨hler manifold H and a Ka¨hler manifold E with parallel Ricci
tensor. Then, any minimal isometric immersion f : M → Rn is totally geodesic.
Since any orientable surface is naturally a Ka¨hler manifold we obtain the following
well-known corollary [2, Proposition 4.1], [10, Chapter IV].
Corollary 1. Let U ⊂ RH2 be an open submanifold of the real hyperbolic
plane RH2. Then, there is no minimal isometric immersion f : U → Rn.
Finally, we obtain the following corollary which is a generalization of the re-
sult in [12] concerning the restricted holonomy group of hypersurfaces in complex
Euclidean spaces.
2000 Mathematics Subject Classification 53B25 (primary), 53C42 (secondary).
Supported by an EPSRC Grant GR/R69174.
2 antonio j. di scala
Corollary 2. Let Mn be a locally irreducible Ka¨hler manifold of complex di-
mension n and assume that there exists a minimal isometric immersion f : M →
R2n+m. Then, the restricted holonomy group of M is U(n). In particular, the re-
stricted holonomy group of an irreducible complex submanifold of Cn+m is U(n).
It interesting to note that a strong version of the last statement of the above
corollary also holds for Ka¨hler manifolds Mn isometrically and holomorphically
immersed in the complex hyperbolic space CHn+p. Namely, a Ka¨hler manifold Mn
isometrically and holomorphically immersed in CHn+p is locally irreducible and its
holonomy group Hol(∇) is U(n) (see [1]).
2. Proof of the main theorem and further remarks
To prove our main theorem we note that Theorem 1.2 from [6] and Theorem 1.11
from [5] implies the following theorem.
Theorem 3. [6] [5] Let M be a simple connected Ka¨hler manifold (not neces-
sary complete) and let f : M → Rn be a minimal isometric immersion. Then there
exist a minimal isometric immersion g : M → Rn such that f : M → Cn = Rn×Rn
is given by f(p) = ( f(p)√
2
, g(p)√
2
) is isometric and holomorphic with respect to the com-
plex structure J(u, v) = (−v, u) on Rn × Rn.
Recall the following remarkable result by M. Umehara.
Theorem 4. [13] Every Ka¨hler-Einstein submanifold of a complex linear or
hyperbolic space is always totally geodesic.
Notice that the above theorem is of local nature. The proof is by using the so
called “diastatic function” introduced by E. Calabi in [4].
We need the following result.
Theorem 5. [7] Let M be an homogeneous Ka¨hler manifold and f : M → Cn
an holomorphic and isometric immersion. Then f is totally geodesic.
Now we can prove our main theorem.
Proof of Theorem 1. Let M˜ be the universal covering of M and f˜ = f ◦pi, where
pi : M˜ → M is the canonical projection. Let f˜ : M˜ → Cn be the isometric and
holomorphic immersion given by Theorem 3. Notice that f is totally geodesic if and
only if f˜ is totally geodesic.
Assume now that M is Ka¨hler-Einstein. Then M˜ is Ka¨hler-Einstein. If p ∈ M˜ is
any point of M˜ then we can restrict f˜ to an small neighborhood U of p ∈ M˜ such
that f˜ |U : U → Cn is an imbedding. Then Theorem 4 implies that f˜ |U is totally
geodesic. So f˜ is totally geodesic.
Assume now that M is homogeneous. Then M˜ is homogeneous. Then by Theorem
minimal immersions of ka¨hler manifolds 3
5 it follows that f˜ is totally geodesic. This completes the proof of our main theorem.
Proof of Corollary 2. Observe that f can not be totally geodesic because we
assume that M is locally irreducible. Theorem 2 shows that the Ricci tensor of M
can not be parallel. Let Φ∗ be the restricted holonomy group of M . The classifi-
cation of irreducible restricted holonomy groups by M. Berger shows that either
Φ∗ is U(n), SU(n), Sp(n) or M is a locally symmetric space [3]. If M is a locally
symmetric space we get that M has parallel Ricci tensor. If Φ∗ is one of the groups
SU(n), Sp(n) we get that M is Ricci-flat. Thus Φ∗ = U(n), as we have to prove.
Let M be a Riemannian manifold and let f : M → Rn be a minimal isometric
immersion. It is interesting to note that f must be totally geodesic if one of the
following conditions holds.
(1) M is complete, the Ricci tensor of M has constant eigenvalues and f has flat
normal bundle (see [8]),
(2) M is Ricci-flat (as follows from Gauss equation),
(3) f has codimension 2 and M is Einstein (see [11]).
So, it seems natural to pose the following conjecture.
Conjecture. Let M be a Riemannian manifold that is either locally homoge-
neous or Einstein. Then, any minimal isometric immersion f : M → Rn must be
totally geodesic.
Acknowledgements. I wish to thank Marcos Dajczer for the helpful discussions
and comments. I also wish to thank Dmitri Alekseevsky and Kevin Baron for read
the manuscript and help me in improving it.
References
1. D.V. Alekseevsky and A. J. Di Scala, ‘The normal holonomy group of Ka¨hler submani-
folds’, Preprint (2002).
2. R. Bryant, ‘Minimal surfaces of constant curvature in Sn’, Trans. A.M.S. 290 (1985) 1
259–271.
3. A.L. Besse, ‘Einstein Manifolds’, Ergebnisse der Mathematik und ihrer Grenzgebiete 3.
Folge - Band 10, Springer Verlag (1987).
4. E. Calabi, ‘Isometric imbedding of complex manifolds’, Ann. of Math. 58 (1953) 1–23.
5. M. Dajczer and D. Gromoll, ‘Real Kaehler submanifolds and uniqueness of the Gauss
map’, J. Differential Geom. 22 (1985) 1 13–28.
6. M. Dajczer and L. Rodriguez, ‘Rigidity of real Kaehler submanifolds’, Duke Math. J. 53
(1986) 1 211–220.
7. A.J. Di Scala, ‘Minimal homogeneous submanifolds in Euclidean spaces’, Ann. Global Anal.
Geom. 21 (2002) 1 15–18.
8. A.J. Di Scala and C. Olmos, ‘Submanifolds with curvature normals of constant length and
the Gauss map’, Preprint (2002).
9. N. Ejiri, ‘Minimal immersions of Riemannian products into real space forms’, Tokyo J. Math.
Vol. 2 Nro. 1 (1979) 63–70.
10. H.B. Lawson, ‘Lectures on minimal submanifolds. Vol. I’, Mathematics Lecture Series, 9.
Berkeley, California: Publish Perish Inc. (1980).
11. Y. Matsuyama, ‘Minimal Einstein submanifolds with codimension two’, Tensor (N.S.) 52
(1993) no. 1 61–68.
4 minimal immersions of ka¨hler manifolds
12. K. Nomizu and B. Smyth, ‘Differential geometry of complex hypersurfaces II’, J. Math. Soc.
Japan 20 (1968) 498–521.
13. M. Umehara, ‘Einstein Ka¨hler submanifolds of a complex linear or hyperbolic space’, Toˆhoku
Math. Journ. 39 (1987) 385–389.
Antonio J. Di Scala
The University of Hull
Cottingham Road
HU6 7RX, Hull, UK
A.J.Di-Scala@maths.hull.ac.uk


Minimal Immersions of Kahler manifolds into Euclidean Spaces

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