Umbilical spacelike submanifolds of arbitrary co-dimension

0.15 MB·

by  Nastassja Cipriani

UMBILICAL SPACELIKE SUBMANIFOLDS
OF ARBITRARY CO-DIMENSION
7 NASTASSJACIPRIANIANDJOSE´ M.M.SENOVILLA
1
0
Abstract. Givenasemi-Riemannianmanifold,wegivenecessaryandsufficientconditions for
2
a Riemannian submanifold of arbitrary co-dimension to be umbilical along normal directions.

b We do that by using the so-called total shear tensor, i.e., the trace-free part of the second
e fundamental form.
F We define the shear space and the umbilical space as the spaces generated by the total shear
tensorandbytheumbilicalvectorfields,respectively.Weshowthatthesumoftheirdimensions
5
mustequaltheco-dimension.
1

]
G
D 1. Introduction
h. The notions of umbilical point and umbilical submanifold are classicalin differential geometry.
t They have mainly been studied in the Riemannian setting and, apart from very few exceptions,
a
m they have been applied to submanifolds of co-dimension one (hypersurfaces). In [1] the authors
studiedtheseconceptsinaslightlymoregeneralframework,namely,allowingtheambientmanifold
[
to have arbitrary signature while requiring the submanifold to be spacelike (i.e.endowed with a

2 Riemannian induced metric). Motivated by the applications to gravitation and general relativity,
v in [1] the authors focused on spacelike submanifolds of co-dimension two. Notice that the results
5
presented in [1] generalized those presented in [6]. In the latter, the study of umbilical surfaces
4
(2-dimensional) in 4-dimensional Lorentzian manifolds was carried out.
0
6 In the present paper we generalize what has been done in [6] and [1]. We consider spacelike
0 submanifolds of arbitrary co-dimension and give a characterization of those which are umbilical
1. with respect to some normal directions. Observe that when the co-dimension is one, the normal
0 bundle is 1-dimensional and thus the submanifold can only be umbilical along the unique normal
7 direction.Ontheotherhand,whentheco-dimensionishigherthanone,thereareseveralpossibil-
1
ities andthe submanifoldcanbe umbilicalwith respectto some normalvectorsbut non-umbilical
:
v with respect to others.
i In order to characterize umbilical spacelike submanifolds we make use of the so-called total
X
shear tensor. This is defined as the trace-freepart of the second fundamental form and it appears
r
a often in the mathematical literature, especially in conformalgeometry. Nevertheless, it had never
been given a name prior to [1] and, surprisingly, its relationship with the umbilical properties of
submanifolds seemed to be almost unknown or is not explicitly mentioned at least.
We introduce the notions of shear space and umbilical space. They are defined as the space
generatedby the image of total shear tensor and the one generatedby the umbilical vector fields,
respectively.They both belong to the normalbundle andthey happen to be mutually orthogonal.
Weshowthattheexistenceofumbilicaldirections“shrinks”theshearspacereducingitsdimension.
Moreprecisely,thedimensions ofthe shearspaceandthe umbilicalspacearelinkedinsuchaway
that the sum of the two must equal the co-dimension of the submanifold. Moreover, in specific
situations –for instance when the ambient manifold is Riemannian– the direct sum of the two
spaces generate the whole normal bundle.
2010 Mathematics Subject Classification. 53B25,53B30,53B50.
Keywords and phrases. Umbilicalpoints,umbilicalsubmanifolds,shear.
ThisworkispartiallysupportedbytheBelgianInteruniversityAttractionPoleP07/18(Dygest)andbytheKU
LeuvenResearchFundproject3E160361“Lagrangianandcalibratedsubmanifolds”.NCandJMMSaresupported
undergrantFIS2014-57956-P(SpanishMINECO–FondosFEDER)andprojectIT956-16oftheBasqueGovernment.
JMMSisalsosupportedunderprojectUFI11/55(UPV/EHU).
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2 N.CIPRIANIANDJ.M.M.SENOVILLA
The plan of the paper is as follows. In Section 2 we recall some basic concepts of submanifold
theory, we introduce the shear objects and give the definitions of umbilical point, umbilical sub-
manifold and umbilical space. In Section 3 we show how the shear space and the umbilical space
arerelated,wepresentnecessaryandsufficientconditionsforthesubmanifoldtobe umbilicaland
give some final remarks.
2. Preliminaries
2.1. Basicconceptsofsubmanifoldtheory. Weconsideranorientablen-dimensionalspacelike
submanifold (S,g) of a semi-Riemannian manifold (M,g¯) with inmersion Φ : S −→ M and co-
dimension k. Hence, g := Φ⋆g¯ is positive definite everywhere on S, so that (S,g) is in particular
an oriented Riemannian manifold. Let X(S) and X(S)⊥ denote the set of tangent and normal
vector fields, respectively, on S. The classical formulas of Gauss and Weingarten provide the
decompositionofthevectorfieldderivativesintotheir tangentandnormalcomponents[3,4, 5]as
∇ Y =∇ Y +h(X,Y), ∀X,Y ∈X(S)
X X
∇ ξ =−A X +∇⊥ξ, ∀ξ ∈X(S)⊥, ∀X ∈X(S)
X ξ X
where ∇ and ∇ are the Levi-Civita connections of (M,g¯) and (S,g) respectively, h is the second
fundamental form orshape tensor oftheimmersionandA theWeingarten operator relativetoξ.
ξ
Thederivation∇⊥ sodefineddeterminesaconnectiononthenormalbundle,h(X,Y)=h(Y,X)∈
X(S)⊥ for all X,Y ∈ X(S) and acts linearly (as a 2-covariant tensor) on its arguments while A
ξ
is self-adjoint for every ξ ∈X(S)⊥. The following relation holds
g(A X,Y)=g¯(h(X,Y),ξ), ∀X,Y ∈X(S), ∀ξ ∈X(S)⊥.
ξ
The mean curvature vector field H ∈ X(S)⊥ is 1/n times the trace (with respect to g) of h, so
that for instance one can write [3, 4, 5]
n
1
H = h(e ,e )
i i
n
Xi=1
where {e1,...,en} denotes an orthonormalframe on X(S).
2.2. The total shear tensor and the shear operators. Using the previous notations and
conventions, the following definition is taken from [1]
Definition 1. Thetotal shear tensor hisdefinedasthetrace-freepartofthesecondfundamental
form:
e
h(X,Y)=h(X,Y)−g(X,Y)H.
The shear operator associated teo ξ ∈ X(S)⊥ is the trace-free part of the corresponding shape
operator:
1
A =A − trA 1
ξ ξ ξ
n
where 1 denotes the identity operator. e e
The total shear tensor and shear operators are obviously related by
(1) g(A X,Y)=g¯(h(X,Y),ξ), ∀X,Y ∈X(S) ∀ξ ∈X(S)⊥.
ξ
To the authors’ kenowledge, theetrace-free part of the second fundamental form had never been
givenanamepriorto[1].Nevertheless,itiseasytofinditintheliterature,inRiemanniansettings,
especiallyinconnectionwiththeconformalpropertiesofsubmanifolds.Apioneeranalysisappears
in [2], where an extensive exposition concerning conformal invariants was given. The total shear
tensor is also in the basis of the definition of the so-called generalized Willmore functional [7].
UMBILICAL SPACELIKE SUBMANIFOLDS OF ARBITRARY CO-DIMENSION 3
Denote by {ξ1,...,ξk} a local frame in X(S)⊥. With respect to this frame, there exist k shear
operators A1,...,Ak such that the total shear tensor h decomposes as
e e k e
(2) h(X,Y)= g(A X,Y)ξ , ∀X,Y ∈X(S).
i i
Xi=1
e e
If{ξ1,...,ξk}isorthonormal,i.e.g¯(ξi,ξj)=ǫiδij withǫ2i =1,thenAi =ǫiAξi foralli.However,in
general,A doesnotneedbeproportionaltoA ,ratherbeingalinearcombinationofA ,...,A .
i ξi e e ξ1 ξk
Given any normal vector field η ∈ X(S)⊥, by (2) its corresponding shear operator A can be
e e e η e
expressed in terms of A1,...,Ak. Indeed, formulas (1) and (2) imply e
e e k
(3) A = g¯(ξ ,η)A .
η i i
Xi=1
e e
In the definitions that follow we introduce two useful concepts.
Definition 2. At any point p∈S, the set
Imh :=span h(v,w):v,w ∈T S ⊆T S⊥
p p p
n o
e e
is called the shear space of S at p.
If N 1 = span{h(v,w):v,w ∈T S} ⊆ T S⊥ denotes the first normal space of S at the point
p p p
p ∈ S, then for every p in S we have Imh ⊆ N 1, hence dimImh ≤ dimN 1 ≤ k. Furthermore,
p p p p
given any orthonormal basis {e1,...,en}ein TpS, the image of hep is spanned by the n(n+1)/2
n
vectors h(e ,e ), for i≤j. Because h(e ,e )=0, these vectors are not linearly independent.
i j i=1 i i e
In particular, the dimension of ImhPcan be at most n(n+1)/2−1. Therefore
e p e
e n(n+1)
(4) dimImh ≤min k, −1 .
p
(cid:26) 2 (cid:27)
e
Formula(3) for the decompositionof any shear operatorimplies that if dimImh =d then any
p
d+1 shear operators must be linearly dependent in p. The converse of this is also true, so that
e
(5) dimImhp =max d|∃η1,...,ηd ∈X(S)⊥ :Aη1,...,Aηd are linearly independent at p
n o
e e e
Notice that similar formulas to (4) and (5) also hold for the dimension of the first normal
space. Indeed, for the dimension of N 1 we will have dimN 1 ≤ min{k,n(n+1)/2}; as for (5),
p p
the Weingarten operators will just take the place of the shear operators. These two formulas for
N 1 imply that if k−n(n+1)/2 is positive, then there exist k−n(n+1)/2 linearly independent
p
Weingarten operators that vanish at p.
Definition 3. Assume that the dimension of the shear spaces Imh is constant on S, i.e.there
p
exists d∈N with 0≤d≤k such that dimImh =d for all p∈S. The set
p e
Imh= eImh ⊆X(S)⊥
p
p[∈S
e e
is calledthe shear space ofS.Then, the shearspace is amodule overthe ringoffunctions defined
on S with dimension d.
The properties already presented relating dimImh to the shear operators can be extended to
p
dimImh accordingly.
e
e
4 N.CIPRIANIANDJ.M.M.SENOVILLA
2.3. Umbilical points and umbilical submanifolds. Forworksconcerningumbilicalsubman-
ifoldsandsomepreviousresultsinbothRiemannianandsemi-Riemanniansettingsthereadercan
consult, e.g., [1] and references therein.
For hypersurfaces (co-dimension 1), a point can only be umbilical along the unique normal
direction. This situation changes completely for higher co-dimensions, in which case there are
multiple directions along which a point can be umbilical.
Definition 4. Using the notations and conventions introduced above for the immersion Φ :
(S,g)→(M,g¯), a point p∈S is said to be
• umbilical with respect to ξ ∈T S⊥ if A is proportionalto the identity;
p p ξp
• totally umbilical if it is umbilical with respect to all ξ ∈T S⊥.
p p
A point p ∈ S is umbilical with respect to ξ ∈ T S⊥ if and only if A = (trA /n)1 or,
p p ξp ξp
equivalently, A =0 for any a∈R\{0}. ξ -umbilicity is thus a property that gives information
aξp p e
aboutspan{ξ }regardlessofthelengthandtheorientationofξ .Hence,wewillusuallystatethat
p p
e
p is umbilical with respect to the normal direction spanned by ξ . On the other hand, p is totally
p
umbilicalif and only if h(v,w)=g(v,w)H for all v,w∈T S or,equivalently, if andonly if h=0
p p
at p. This fact was already known for hypersurfaces in Riemannian settings and can be found,
e
e.g., in [2]. However, the relationship between the total shear tensor and the umbilical properties
of submanifolds, treated in [1] and in the present article, is substantially new.
Definition 5. Given any point p∈S, the set
U = ξ ∈T S⊥ :p is umbilical with respect to ξ ⊆T S⊥
p p p p p
(cid:8) (cid:9)
is called the umbilical space of S at p.
Lemma 1. The umbilical space U is a vector space for every p∈S.
p
Proof. Let ξ ,η ∈U , so that by definition A =A =0. Let a,b∈R and consider the normal
p p p ξp ηp
vector aξp+bηp. By linearity we have Aaξp+beηp =aeAξp +bAηp =0. It follows that p is umbilical
with respect to aξ +bη , hence aξ +bη belongs to U . (cid:3)
p p p p p
e e e
It followsfrom Lemma 1 that dimU is welldefined. Notice that dimU =m if andonly if p is
p p
umbilicalwithrespecttoexactly mlinearlyindependentnormaldirections.Moreover,byformulas
(4) and (5) it follows that if k−n(n+1)/2+1 is positive, then dimU ≥k−n(n+1)/2+1.
p
Definition 6. Using the notations and conventions introduced above for the immersion Φ :
(S,g)→(M,g¯), the submanifold (S,g) is said to be
• umbilical with respect to ξ ∈X(S)⊥ if A is proportionalto the identity;
ξ
• totally umbilical if it is umbilical with respect to all ξ ∈X(S)⊥.
The propertiespresentedabovefor umbilicalpoints canbe extendedto umbilicalsubmanifolds
accordingly. S is umbilical with respect to ξ ∈ X(S)⊥ if and only if ξ ∈ U for all p ∈ S. More
p p
in general, S is umbilical with respect to exactly m linearly independent non-zero normal vector
fields ξ1,...,ξm ∈ X(S)⊥ if and only if (ξ1)p,...,(ξm)p ∈ Up for all p ∈ S. Equivalently, if and
only if dimU =m for all p∈S. This leads to the following definition.
p
Definition 7. Assume that the dimension of the umbilical spaces U is constant on S, i.e.there
p
exists m∈N with 0≤m≤k such that dimU =m for all p∈S. Then the set
p
U = ξ ∈X(S)⊥ :S is umbilical with respect to ξ ⊆X(S)⊥
(cid:8) (cid:9)
is called the umbilical space of S.
The umbilicalspace ofS is suchthatU =∪p∈SUp. InLemma 1 we provedthat Up is a vector
space for every p. Similarly we can prove that U is a finitely generated module over the ring of
functions defined on S with dimU =m.
UMBILICAL SPACELIKE SUBMANIFOLDS OF ARBITRARY CO-DIMENSION 5
3. Results
3.1. The relationship between U and Imh.
Proposition 1. Let Φ : (S,g) → (M,g¯) be ean isometric immersion of an n-dimensional Rie-
mannian manifold into a semi-Riemannian manifold with co-dimension k. Let U and Imh be
p p
the umbilical space and the shear space, respectively, of S at any point p∈S. Then
e
U =(Imh )⊥.
p p
Moreover, e
k−dimU =dimImh .
p p
Here (Imh )⊥ is defined as the subspace of T S⊥ orthoegonalto Imh , namely
p p p
e (Imh )⊥ = η ∈T S⊥|∀ξ ∈Imh :g¯(η ,ξ )=e0 .
p p p p p p p
n o
e e
Proof. Bydefinition,anormalvectorξ belongstoU ifA =0.Equivalently,ifg(A (v),w)=0
p p ξp ξp
inp,forallv,w ∈T S.Byformula(1),thisholdsifandonlyifξ ∈(Imh )⊥.HenceU =(Imh )⊥.
p e p p ep p
Suppose Imh = {0}, then it is clear that U = (Imh )⊥ = T S⊥ and the relation between the
p p p p e e
dimensionsheolds.NowassumeImhp 6={0}.Wecancheooseabasis{(ξ1)p,...,(ξk)p}ofTpS⊥ such
that {(ξ1)p,...,(ξd)p} is a basis oef Imhp. A normal vector ηp belongs to Up = (Imhp)⊥ if and
only if g¯(ηp,(ξj)p) = 0 for j = 1,...,de. Since (ξ1)p,...,(ξd)p are linearly independeent and g¯ is
non-degenerate, these are d linearly independent conditions on the components of η and hence
p
dimU =k−d. (cid:3)
p
By Proposition 1 we have that the umbilical space U and the shear space Imh are such that
p p
U =(Imh )⊥ and dimU +dimImh =k. e
p p p p
However,the intersectionU ∩Imeh might be non-empty,andconseequently the directsum of the
p p
two spaces does not generate, in general, the whole normal space. For example, if p is umbilical
e
with respect to some vector ξ that is null, i.e. g¯(ξ ,ξ ) = 0, then ξ might belong to Imh .
p p p p p
Actually, in caseM is a Riemannian manifold,one easily checksthat Imh ∩U =∅ andone has
p p e
T S⊥ =Imh ⊕U (M Riemannian)e.
p p p
Remark 1. Proposition1implicitlyeshowsthatifthedimensionoftheshearspacesImh isconstant
p
on S then the dimension of the umbilical spaces U also is, and vice versa.
p
e
By Proposition 1 and Remark 1 follows the next corollary.
Corollary 1. Assume that the dimension of the shear spaces Imh is constant on S (equivalently,
p
that the dimension of the umbilical spaces U is constant on S). Then Imh and U are well defined
p e
and we have
e
U =(Imh)⊥.
Moreover, e
k−dimU =dimImh.
From now on, and for the sake of conciseness, we will aessume that the dimension of the shear
spaces Imh is constant on S. We will consider that S is umbilical with respect to exactly m
p
linearly independent umbilical directions, that is to say, dimU = m. Notice, however, that all
e
results make sense also if stated pointwise.
6 N.CIPRIANIANDJ.M.M.SENOVILLA
3.2. Characterization of umbilical spacelike submanifolds. We willdenoteby ∧the wedge
product of one-forms and by ♭ the musical isomorphism: if V is a vector field on (M,g¯), then its
associated one-form V♭ is given by V♭(Z)=g¯(V,Z) for every vector field Z on M.
Proposition 2. Let Φ : (S,g) → (M,g¯) be an isometric immersion of an n-dimensional Rie-
mannian manifold into a semi-Riemannian manifold with co-dimension k. Let U be the umbilical
space of S, then dimU =m if and only if the total shear tensor satisfies
k−m+1
h♭ =0
^
e
with
k−m
h♭ 6=0.
^
q e
Here by ω we mean q times the wedge product ω∧···∧ω of any one form ω. Notice that
whenwewrVite,forinstance,h♭∧h♭,wemeanh♭(X1,Y1)∧h♭(X2,Y2)forallX1,Y1,X2,Y2 ∈X(S).
e e e e
Proof. Suppose that the dimension of the umbilical space is m. By Corollary 1 it follows that
dimImh = k−m and we can then decompose the total shear tensor by means of exactly k−m
normalvectorfields.Explicitly,thereexistk−mshearoperators{A }k−m andk−mvectorfields
e i i=1
ζ1,...,ζk−m ∈X(S)⊥ such that e
k−m
h(X,Y)= g(A X,Y)ζ .
i i
Xi=1
e e
Because these vector fields are linearly independent, their corresponding one-forms ζ♭ also are.
r
Fromthis factiteasilyfollowsthat,ononehand,the wedgeproductk−mtimes ofh♭ is different
from zero and, on the other hand, that the wedge product k−m+1 times of h♭ must be zero.
e
Now suppose that the total shear tensor satisfies the conditions in the statement. By algebra’s
e
basicresults,weknowthatlone-formsω1,...,ωlarelinearlyindependentifandonlyiftheirwedge
productω1∧···∧ωl isnotzero.Equivalently,theyarelinearlydependentifandonlyiftheirwedge
productis zero.Hence, by hypothesis,amongthe sets ofk one-forms{h(X1,Y1)♭,...,h(Xk,Yk)♭}
constructed for arbitrary X1,Y1,...,Xk,Yk ∈ X(S), there exist exactley k−m whicheare linearly
independent.Thesameholdsforthecorrespondingnormalvectorfields{h(X1,Y1),...,h(Xk,Yk)}.
ByDefinition2ofshearspace,thisimpliesthatthedimensionofImhisk−m.FinallybyCorollary
e e
1, we obtain dimU =m. (cid:3)
e
The following theorem summarizes the results presented in Corollary 1 and Proposition 2.
Theorem1. LetΦ:(S,g)→(M,g¯)beanisometricimmersionofann-dimensionalRiemannian
manifold into a semi-Riemannian manifold with co-dimension k. Let U and Imh be the umbilical
space and the shear space, respectively, of S. Then the following conditions are all equivalent:
e
(i) the umbilical space U has dimension m;
(ii) the shear space Imh has dimension k−m;
(iii) the total shear tensor satisfies
e
k−m+1
h♭ =0
^
e
with
k−m
h♭ 6=0;
^
e
(iv) any k −m+1 shear operators A ,...,A are linearly dependent (and there exist
ξ1 ξk−m+1
precisely k−m shear operators that are linearly independent).
e e
UMBILICAL SPACELIKE SUBMANIFOLDS OF ARBITRARY CO-DIMENSION 7
3.3. Special cases. Using Theorem 1 some particular situations are worth mentioning, for ex-
ample:
• If dimU = k we have h(X,Y)♭ = 0 for every X,Y ∈ X(S), equivalently h = 0, and the
submanifold S is totally umbilical. In particular, Imh=∅ and X(S)⊥ =U.
e e
• If dimU = k−1 then dimImh = 1 and we have he(X1,Y1)♭ ∧h(X2,Y2)♭ = 0 for every
X1,Y1,X2,Y2 ∈ X(S). It follows that there exist a normal vector field G ∈ X(S)⊥ and
e e e
a properly normalized self-adjoint operator A such that h(X,Y) = g(AX,Y)G for every
X,Y ∈X(S). This was the case studied in [1] for k =2.
e e e
• If dimU =0 then there are no umbilical directions and Imh=X(S)⊥.
Our results can be applied to all other cases too and open the door for a novel analysis of the
e
structure of the umbilical space.
Acknowledgement:We would like tothank thereferee for comments and improvements.
References
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KULeuven,DepartmentofMathematics,Celestijnenlaan200B–Box2400,BE-3001Leuven,Belgium
andF´ısica Teo´rica, Universidaddel Pa´ıs Vasco, Apartado644,48080Bilbao,Spain
E-mail address: [email protected]
F´ısica Teo´rica,Universidad del Pa´ısVasco, Apartado644,48080Bilbao,Spain
E-mail address: [email protected]