The object of this article is to characterize submanifolds of the Euclidean space whose shape operator is unipoten

Di Scala, Antonio Jose'

English

PORTO Publications Open Repository TOrino

On submanifolds whose shape operator is unipotent
Antonio J. Di Scala
Post print (i.e. final draft post-refereeing) version of an article published on J. Geom. 85 (2006), no. 1-2, 3234. 10.1007/s00022-
006-0039-z .
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://rd.springer.com/article/10.1007%2Fs00022-006-0039-z
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/issn/0047-2468/
Abstract
The object of this article is to characterize submanifolds M ⊂ Rn of
the Euclidean space whose shape operator Aξ satisfies the equation (Aξ)2 =
k‖ξ‖2Id, where k > 0 is constant.
Mathematics Subject Classification(2000): 53C42, 53B25
Key Words: unipotent shape operator, constant principal curvatures, normal
holonomy group.
1 Introduction.
Characterization of submanifolds of the Euclidean space by imposing conditions
on its extrinsic or intrinsic invariants is a classical subject of the differential geom-
etry ([7], [2]) e.g. minimal submanifolds, isometric immersions of real space forms,
isoparametric submanifolds, etc.
The shape operator Aξ of a submanifold M ⊂ M˜ is called unipotent if it satisfies
the equation
(Aξ)2 = k‖ξ‖2Id,
where ξ belongs to the normal bundle T⊥M of the submanifold M . The following
theorem was proved in [1, Theorem 14].
Theorem Let Mm be a Ka¨hler submanifold of the complex space form Snc =
CPn,Cn,CHn (n > m > 1) with c = 4, 0,−4 and Aξ its shape operator. Then, the
following conditions are equivalent:
1
2 Antonio J. Di Scala
(i) (Aξ)2 = f‖ξ‖2Id, where f is a positive function on M ,
(ii) c = 4 and M is an open subset of the complex quadric Qm := {[z0 : · · · :
zm+1] ∈ CPm+1 | z20 + · · ·+z2m+1 = 0} . In particular f = ( 4c )2 is a constant.
The object of this article is to give a similar characterization for submanifolds of
the Euclidean space. We get the following result.
Theorem 1.1 Let Mm ⊂ Rn be a submanifold of the Euclidean space and let Aξ
its shape operator. Then, the following facts are equivalent:
(i) (Aξ)2 = k‖ξ‖2Id, where k > 0 is constant;
(ii) Mm is an open subset of a hypersphere or an open subset of the Veronese
embedding of the real, complex, quaternionic or Cayley projective planes in the
sphere S3m+1 where m = 1, 2, 4, 8, i.e. a focal manifold of an isoparametric
hypersurface of a sphere with three distinct principal curvatures.
It is interesting to remark that the proof uses techniques related with the normal
holonomy group of the given submanifold. Normal holonomy groups were classi-
fied by C. Olmos [6] and they are very useful for the study of the geometry of
homogeneous submanifolds [2].
2 Submanifolds with constant principal curvatures.
A submanifold M ⊂ Rn has constant principal curvatures if the shape operator
Aξ(t) has constant eigenvalues for any∇⊥-parallel normal vector field ξ(t) along any
piecewise differentiable curve. If in addition the normal bundle is flat i.e. R⊥ ≡ 0
then M is called isoparametric [7]. We will need the following result.
Theorem 2.1 [5] A submanifold M of Euclidean space has constant principal cur-
vatures if and only if is either isoparametric or a focal manifold to an isoparametric
submanifold.
For a discussion of focal manifolds of an isoparametric submanifold, its related
Coxeter group, etc, see [2, Chapter 5].
On submanifolds whose shape operator is unipotent 3
3 Proof of Theorem 1.1.
It is well-known that (ii) implies (i) (see [3] or [4] for details).
Before dwelling into the proof that (i) implies (ii), we briefly explain the main
ideas. In first place we will show that the submanifold M has principal constant
curvatures. Then, from Theorem 2.1 there exists an isoparametric submanifold N
such that M = Nξ i.e. M is a parallel submanifold of N . Then we will show that
N is a hypersurface of a hypersphere. Since N is a hypersurface it follows that M
is obtained from N by focalizing an eigendistribution i.e. M = N ξ
λ
for a constant
principal curvature λ of N , see [2, pag.119, Example 4.2].
Proof of Theorem 1.1. Since Aξ(t) satisfies a polynomial equation then the shape
operator Aξ(t) with respect to any ∇⊥-parallel normal section ξ(t) has constant
eingenvalues. So the submanifold has constant principal curvatures.
Then, as a consequence of Theorem 2.1, either M is isoparametric or is a focal
manifold of an isoparametric submanifold. If M is isoparametric then it is not dif-
ficult to see that M is a piece of an hypersphere.
Assume now that M is a focal submanifold of an isoparametric submanifold N .
Then from the proof of Theorem 2.1 it follows that N is a holonomy tube (M)ξp
through a principal vector ξp ∈ T⊥p M . We claim that N is a isoparametric hyper-
surface of a hypersphere. In fact, from [2, pag.126, Remark 4.4.13] it is enough to
show that the codimension of a principal orbit Holp(∇⊥).ξp is one. Indeed, if it is
greater than one, we will obtain a contradiction. Note first that the Ricci equation
implies [Aξ, Aη] = 0 for any two normal vectors ξ, η to a principal orbit of the action
of the normal holonomy group. So, we can perform a simultaneous diagonalization
of the family {Aξ : ξ ∈ the normal space to a principal orbit }. Let λ(ξ) = 〈r, ξ〉
be a common eigenvalue. Thus, if η ⊥ r is in the normal space of the principal
orbit then (Aη)2 6= kId and we get a contradiction.
Thus, N is a isoparametric hypersurface of a hypersphere and the submanifold
M is a focal submanifold of N i.e. M = N ξ
λ
where ξ is the unit normal of N
and λ a constant principal curvature of N . Since the shape operator Aξ of M has
two eigenspaces for any normal vector ξ then the “Tube formula” (see [2, pag.121,
Lemma 4.4.7]) implies that the shape operator of N must have three principal cur-
vatures.
So from a theorem of Elie Cartan [3] it follows that M must be one of the cited
embeddings (for a beautiful proof of this Cartan’s Theorem see [4]). 2
References
[1] Alekseevsky, D.V. and Di Scala, A.J.: The normal holonomy group of Ka¨hler
4 Antonio J. Di Scala
submanifolds, Proc. London Math. Soc. (3) 89 (2004) 193-216.
[2] Berndt, J. ; Console, S. and Olmos, C.: Submanifolds and holonomy, Research
Notes in Mathematics 434, Chapman & Hall/CRC, 2003.
[3] Cartan, E.: Sur des familles remarquables d’hypersurface isoparame´triques dans les
espaces sphe´riques, Math. Z. 45 (1939) 335-367.
[4] Console, S. and Olmos, C.: Clifford systems, algebraically constant second funda-
mental form and isoparametric hypersurfaces Manuscripta Math. 97 (1998) 335-342.
[5] Heintze, E. ; Olmos, C. and Thorbergsson G.: Submanifolds with constant prin-
cipal curvatures and normal holonomy groups, Internat. J. Math. 2 (1991) 167-175.
[6] Olmos, C.: The normal holonomy group, Proc. Amer. Math. Soc. 110 (1990) 813-818.
[7] Palais, R. and Terng, C.-L.: Critical point theory and submanifold geometry, Lec-
tures Notes in Math., vol. 1353, Springer-Verlag, Berlin, 1988.
Dipartimento di Matematica, Politecnico di Torino,
Corso Duca degli Abruzzi 24, 10129 Torino, Italy.
antonio.discala@polito.it


A.J.: The normal holonomy group of K ahler4 Antonio J. Di Scala submanifolds,

Submanifolds and holonomy,

On submanifolds whose shape operator is unipotent

Antonio J. Di Scala

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The object of this article is to characterize submanifolds
of the Euclidean space whose shape operator is unipotent

DI SCALA, ANTONIO JOSE'

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

09 November 2022POLITECNICO DI TORINORepository ISTITUZIONALEOn submanifolds whose shape operator is unipotent / DI SCALA, ANTONIO JOSE'. - In: JOURNAL OF GEOMETRY. -ISSN 0047-2468. - STAMPA. - 85:(2006), pp. 32-34. [10.1007/s00022-006-0039-z]OriginalOn submanifolds whose shape operator is unipotentPublisher:PublishedDOI:10.1007/s00022-006-0039-zTerms 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: 11583/1399750 since: 2015-12-01T11:10:57ZBirkhäuser-VerlagOn submanifolds whose shape operator is unipotentAntonio J. Di ScalaPost print (i.e. final draft post-refereeing) version of an article published on J. Geom. 85 (2006), no. 1-2, 3234. 10.1007/s00022-006-0039-z .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://rd.springer.com/article/10.1007%2Fs00022-006-0039-zThis 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/issn/0047-2468/AbstractThe object of this article is to characterize submanifolds M ⊂ Rn ofthe Euclidean space whose shape operator Aξ satisfies the equation (Aξ)2 =k‖ξ‖2Id, where k > 0 is constant.Mathematics Subject Classification(2000): 53C42, 53B25Key Words: unipotent shape operator, constant principal curvatures, normalholonomy group.1 Introduction.Characterization of submanifolds of the Euclidean space by imposing conditionson its extrinsic or intrinsic invariants is a classical subject of the differential geom-etry ([7], [2]) e.g. minimal submanifolds, isometric immersions of real space forms,isoparametric submanifolds, etc.The shape operator Aξ of a submanifold M ⊂ M̃ is called unipotent if it satisfiesthe equation(Aξ)2 = k‖ξ‖2Id,where ξ belongs to the normal bundle T⊥M of the submanifold M . The followingtheorem was proved in [1, Theorem 14].Theorem Let Mm be a Kähler submanifold of the complex space form Snc =CPn,Cn,CHn (n > m > 1) with c = 4, 0,−4 and Aξ its shape operator. Then, thefollowing conditions are equivalent:12 Antonio J. Di Scala(i) (Aξ)2 = f‖ξ‖2Id, where f is a positive function on M ,(ii) c = 4 and M is an open subset of the complex quadric Qm := {[z0 : · · · :zm+1] ∈ CPm+1 | z20 + · · ·+z2m+1 = 0} . In particular f = ( 4c )2 is a constant.The object of this article is to give a similar characterization for submanifolds ofthe Euclidean space. We get the following result.Theorem 1.1 Let Mm ⊂ Rn be a submanifold of the Euclidean space and let Aξits shape operator. Then, the following facts are equivalent:(i) (Aξ)2 = k‖ξ‖2Id, where k > 0 is constant;(ii) Mm is an open subset of a hypersphere or an open subset of the Veroneseembedding of the real, complex, quaternionic or Cayley projective planes in thesphere S3m+1 where m = 1, 2, 4, 8, i.e. a focal manifold of an isoparametrichypersurface of a sphere with three distinct principal curvatures.It is interesting to remark that the proof uses techniques related with the normalholonomy group of the given submanifold. Normal holonomy groups were classi-fied by C. Olmos [6] and they are very useful for the study of the geometry ofhomogeneous submanifolds [2].2 Submanifolds with constant principal curvatures.A submanifold M ⊂ Rn has constant principal curvatures if the shape operatorAξ(t) has constant eigenvalues for any∇⊥-parallel normal vector field ξ(t) along anypiecewise differentiable curve. If in addition the normal bundle is flat i.e. R⊥ ≡ 0then M is called isoparametric [7]. We will need the following result.Theorem 2.1 [5] A submanifold M of Euclidean space has constant principal cur-vatures if and only if is either isoparametric or a focal manifold to an isoparametricsubmanifold.For a discussion of focal manifolds of an isoparametric submanifold, its relatedCoxeter group, etc, see [2, Chapter 5].On submanifolds whose shape operator is unipotent 33 Proof of Theorem 1.1.It is well-known that (ii) implies (i) (see [3] or [4] for details).Before dwelling into the proof that (i) implies (ii), we briefly explain the mainideas. In first place we will show that the submanifold M has principal constantcurvatures. Then, from Theorem 2.1 there exists an isoparametric submanifold Nsuch that M = Nξ i.e. M is a parallel submanifold of N . Then we will show thatN is a hypersurface of a hypersphere. Since N is a hypersurface it follows that Mis obtained from N by focalizing an eigendistribution i.e. M = N ξλfor a constantprincipal curvature λ of N , see [2, pag.119, Example 4.2].Proof of Theorem 1.1. Since Aξ(t) satisfies a polynomial equation then the shapeoperator Aξ(t) with respect to any ∇⊥-parallel normal section ξ(t) has constanteingenvalues. So the submanifold has constant principal curvatures.Then, as a consequence of Theorem 2.1, either M is isoparametric or is a focalmanifold of an isoparametric submanifold. If M is isoparametric then it is not dif-ficult to see that M is a piece of an hypersphere.Assume now that M is a focal submanifold of an isoparametric submanifold N .Then from the proof of Theorem 2.1 it follows that N is a holonomy tube (M)ξpthrough a principal vector ξp ∈ T⊥p M . We claim that N is a isoparametric hyper-surface of a hypersphere. In fact, from [2, pag.126, Remark 4.4.13] it is enough toshow that the codimension of a principal orbit Holp(∇⊥).ξp is one. Indeed, if it isgreater than one, we will obtain a contradiction. Note first that the Ricci equationimplies [Aξ, Aη] = 0 for any two normal vectors ξ, η to a principal orbit of the actionof the normal holonomy group. So, we can perform a simultaneous diagonalizationof the family {Aξ : ξ ∈ the normal space to a principal orbit }. Let λ(ξ) = 〈r, ξ〉be a common eigenvalue. Thus, if η ⊥ r is in the normal space of the principalorbit then (Aη)2 6= kId and we get a contradiction.Thus, N is a isoparametric hypersurface of a hypersphere and the submanifoldM is a focal submanifold of N i.e. M = N ξλwhere ξ is the unit normal of Nand λ a constant principal curvature of N . Since the shape operator Aξ of M hastwo eigenspaces for any normal vector ξ then the “Tube formula” (see [2, pag.121,Lemma 4.4.7]) implies that the shape operator of N must have three principal cur-vatures.So from a theorem of Elie Cartan [3] it follows that M must be one of the citedembeddings (for a beautiful proof of this Cartan’s Theorem see [4]). 2References[1] Alekseevsky, D.V. and Di Scala, A.J.: The normal holonomy group of Kähler4 Antonio J. Di Scalasubmanifolds, Proc. London Math. Soc. (3) 89 (2004) 193-216.[2] Berndt, J. ; Console, S. and Olmos, C.: Submanifolds and holonomy, ResearchNotes in Mathematics 434, Chapman & Hall/CRC, 2003.[3] Cartan, E.: Sur des familles remarquables d’hypersurface isoparamétriques dans lesespaces sphériques, Math. Z. 45 (1939) 335-367.[4] Console, S. and Olmos, C.: Clifford systems, algebraically constant second funda-mental form and isoparametric hypersurfaces Manuscripta Math. 97 (1998) 335-342.[5] Heintze, E. ; Olmos, C. and Thorbergsson G.: Submanifolds with constant prin-cipal curvatures and normal holonomy groups, Internat. J. Math. 2 (1991) 167-175.[6] Olmos, C.: The normal holonomy group, Proc. Amer. Math. Soc. 110 (1990) 813-818.[7] Palais, R. and Terng, C.-L.: Critical point theory and submanifold geometry, Lec-tures Notes in Math., vol. 1353, Springer-Verlag, Berlin, 1988.Dipartimento di Matematica, Politecnico di Torino,Corso Duca degli Abruzzi 24, 10129 Torino, Italy.antonio.discala@polito.it

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