On parallel submanifolds · Parallel totally real submanifolds RN = Cn – the complex aﬃne space...

On parallel submanifolds

Barbara Opozda

Leuven, August 2012

Barbara Opozda () On parallel submanifolds Leuven, August 2012 1 / 42

The talk includes a part of the paper

B.Opozda, U. Simon Parallel hypersurfaces (in preparation)


Parallelism, equivalence modulo a group

The ambient space will be the affine space RN .

Definition

Two immersions x1, x2 : M → RN are parallel if

x1∗(TpM) = x2∗(TpM)

for each point p ∈ M.

The immersions x1, x2 are parallel if and only if there is a (1, 1)-tensor fieldL on M such that

dx2 = dx1 L.

The tensor field L will be called the deformation tensor from x1 to x2.It is necessarily non-singular.


The immersions x1, x2 are equivalent modulo a group G ⊂ GA(N,R) if

x2 = Ax1

for some A ∈ G . If A is a translation, then we say that x1, x2 differ bytranslation. Two immersions differ by translation if and only if they areparallel and their deformation tensor is the identity. If L = λ id , whereλ ∈ R \ 0, then x1, x2 are affine homothetic, that is,

x2 = λ x1 + C

for some C ∈ RN . The homothety is the trivial case of parallelism.


Two parallel submanifolds are usually very much different from each other.For instance, any non-degenerate hypersurface of the Euclidean space RNis parallel to its Gauss map (in this case – an immersion), which isspherical, hence totally umbilical.The questions under consideration:

Are there properties which are preserved by the parallelism within somecategories of submanifolds?

Find examples of properties which imply new properties on some parallelsubmanifols.

The answer depends on1) whether the ambient space is regarded as a Euclidean (orpseudo-Euclidean) space or an affine space2) what type of submanifolds are taken into account (e.g. theircodimension)3) which structure on submanifolds we consider (which transversal bundlewe choose)


x : M → RN – an immersionN – a smooth transversal bundle for xthe Gauss formula:

(X (Yx)) = x∗(∇XY ) + h(X ,Y ),

x∗(∇XY ) – the tangential part of (X (Yx))h(X ,Y ) – the transversal bundle part of (X (Yx))∇ – a torsion-free connection on M, the induced connectionh – an RN -valued 2-covariant symmetric tensor field on M, the secondfundamental form (RN – valued) for (x ,N )


L – a non-singular (1, 1)-tensor field on M

Fact

L is a deformation tensor for some immersion x : M → RN if and only ifthe RN -valued form dx L is exact.

Theorem

Let x : M → RN be an immersion with a non-singular (1, 1)-tensor field L.The form dx L is closed if and only if ∇L(·, ·) and h(L·, ·) are symmetric.If dx L is closed and M is simply connected, there is x : M → RN parallelto x with deformation tensor L.x is unique up to translation.


x , x : M → RN – parallel immersions with the deformation tensor L from xto x .Endow both immersions with the same transversal vector bundle N .Denote by ∇, h; ∇, h the induced objects respectively. We have

∇ = L−1∇L = ∇+ L−1(∇L), (1)

h(X ,Y ) = h(LX ,Y ). (2)

Hence L−1(∇L) = ∇ − ∇.

Fact∇L(·, ·) is symmetric and h(L·, ·) is symmetric.

Fact

∇ = ∇ if and only if ∇L = 0.

FactIf M is connected and the deformation tensor L is proportional to theidentity by a function, then the function must be constant.


N.Hicks: Linear perturbations of connections, Mich. Math. J. 12 (1965)


Parallel hypersurfaces

x : M → Rn+1 a hypersurface, that is dimM = n and x is an immersion,M connectedN – a transversal bundle for xξ – a transversal vector field (maybe local) – a section of NThe fundamental formulas:

X (Yx) = dx(∇XY ) + h(X ,Y )ξ −− Gauss formula

X ξ = −dx(SX ) + τ(X )ξ −− Weingarten formula

A transversal vector field ξ is equiaffine if τ = 0 on M.A transversal bundle is equiaffine if it admits equiaffine sections.Two equiaffine transversal vector fields (defined on the same region),determining the same transversal bundle, differ by a constant.A transversal bundle is equiaffine if and only if dτ = 0, where τ is givenfor an arbitrary (not necessarily equiaffine) section ξ of N .


Assume that x : M → Rn+1 is equipped with an equiaffine transversalvector field ξ. The following fundamental equations are satisfied:

R(X ,Y )Z = h(Y ,Z )SX − h(X ,Z )SY

∇h(X ,Y ,Z ) = ∇h(Y ,X ,Z )

∇S(X ,Y ) = ∇S(Y ,X )

h(X ,SY ) = h(SX ,Y )

Fact

If for a hypersurface x : M → Rn+1 with an equiaffine transversal vectorfield ξ the shape operator S is non-singular, then ξ : M → Rn+1 is animmersion parallel to x .


x , x : M → Rn+1 – parallelL – the deformation tensor from x to xξ, ξ – transversal vector fields for x and x respectively

ξ = φξ + x∗(Z ) = φξ + x∗(LZ ), (3)

φ – a nowhere vanishing function, Z is a tangent vector field on M

Formulas for parallel immersions with possibly different transversalvector bundles

τ(X )− τ(X ) = d(log |φ|)(X ) + 1φh(X , LZ ), (4)

LSX = φSX + τ(X )LZ −∇XLZ , (5)

∇XY = L−1∇XLY − h(X ,Y )Z = ∇XY +L−1∇L(X ,Y )− h(X ,Y )Z , (6)

h(X ,Y ) =1φh(X , LY ) (7)

for X ,Y tangent to M.Barbara Opozda () On parallel submanifolds Leuven, August 2012 12 / 42

Z = 0 ⇒ x , x are parallel with the same transversal bundles

x , x parallel with a transversal bundle N ⇒N is equiaffine for x if and only if it is equiaffine for x

The equaiffinity of a transversal bundle is a property of the vector bundlex∗(TM), i.e. the property is independent of a particular mapping x withina class of parallel immersions of M.

Parallel immersions are simultaneously non-degenerate or degenerate.A similar statement does not hold for locally strongly convex hypersurfaces.


Proposition

Let x , x : M → Rn+1 be parallel hypersurfaces with a transversal bundleN . Then

LS = φS , (8)

∇ = L−1∇L. (9)

If S is proportional to S by a nowhere vanishing function and S isnon-singular, then f and f are affine homothetic.

Proposition

Let x , x : M → Rn+1 be non-degenerate parallel hypersurfaces. If they areaffine equivalent, then they are affine homothetic.


Theorem

Let x , x : M → Rn+1, n > 2, be non-degenerate parallel immersions withthe deformation tensor L from x to x and ξ be a transversal vector field tox inducing the shape operator S . If rkS > 1 on M and ∇L = 0, where ∇is the connection induced by ξ, then x and x are affine homothetic.

CorollaryLet x , x be parallel non-degenerate hypersurfaces in the Euclidean spaceRn+1, n > 2. Let L be the deformation tensor from x to x . If ∇L = 0,where ∇ is the induced Levi-Civita connection for x , then x , x are affinehomothetic.


In some cases hypersurfaces are naturally equipped with a transversalbundle. For instance, Euclidean or pseudo-Euclidean hypersurfaces, orBlaschke hypersurfaces.For parallel hypersurfaces in the Euclidean space RN the Euclidean normalbundles and the Euclidean normal vector fields are automatically the same.For parallel hypersurfaces equipped with the Blaschke sructure (shortlyBlaschke hypersurfaces) the affine normal bundles are usually different. Wehave

Proposition

Let x , x : M → Rn+1 be parallel Blaschke hypersurfaces with deformationtensor L. They have the same affine normal bundle if and only if det L isconstant. In such a case the affine normal vector fields are related byξ = ϕξ where

ϕ = ε|det L|−1n+2

and where ε = sign det L.


If for a Blaschke hypersurface the affine shape operator S is non-singular,then the affine normal ξ is an immersion parallel to x and the deformationtensor from x to ξ is equal to S . If detS is constant, then the affinenormal immersion ξ has the same affine normal bundle as x hence ξ is aaffine sphere.

Proposition

If x : M → Rn+1 is a Blaschke hypersurface and the affine Gauss curvaturedetS is a non-zero constant, then x is parallel to a proper affine sphere.

x , x : M → Rn+1 parallel, x - affine sphere ; x affine spherex is an improper affine sphere, x is parallel to x and det L = constant⇒ x is an improper affine sphere (usually non-homothetic to x).


Parallel totally real submanifolds

RN = Cn – the complex affine space with the standard complex structureJ, the standard differentiation and the canonical complex determinantdet CM – n-dimensional real manifold, connected, orientedx : M → Cn – a totally real submanifold, that is, x is an immersion suchthat Jx∗(TpM) is transversal to x∗(TpM) for each p ∈ M, equivalentlyΩ = f ∗(det C) vanishes nowhere on MΩ is a complex-valued real n–form on M(X1, ...,Xn) – a positively oriented basis of TpMΩ(X1, ...,Xn) = µeiθ

µ – a positive real number and θ ∈ Reiθ – independent of a choice of a basis X1, ...,Xnθ – the phase function of xτ = dθ is a globally defined 1-form on M, the Maslov form

Factx : M → Cn minimal if and only if τ = 0 (the phase function is constant)


X (Yx) = x∗(∇XY ) + Jx∗Q(X ,Y ) (10)

where ∇XY and Q(X ,Y ) are tangential to x∇ – a torsion-free connection, the induced connectionQ – a symmetric (1, 2)–tensor field, the second fundamental tensorQXY = Q(X ,Y )

Fact

τ(X ) = trQX

x is minimal if and only if trQX = 0 for every X .

Propositionx , x : M → Cn totally real, parallel with the deformation tensor L. Then

QX = L−1QXL.

The phase functions for both immersions are the same. In particular, x isminimal if and only if x is minimal.


One-parameter familes of parallel hypersurfaces

x : M → Rn+1 – immersion, ξ an equiaffine transversal vector field.The family

xt = x + tξ

will be called a one-parameter parallel family. In general, xt does nothave to be an immersion on the whole M. It can happen that for some tthe image of xt reduces to a point. It is also not true that for some small tthe mappings xt are immersions. If xt is an immersion then it is parallel tox with deformation tensor

Lt = id− tS

and ξ is an equiaffine transversal field for xt . In order that xt is animmersion on M it is sufficient and necessary that det (id− tS) 6= 0 ateach point of M.If xt is an immersion (we shall say that t is admissible), then we equip xtwith the equiaffine transversal vector field ξ.


If the shape operator for x is diagonalizable, then the shape operator

St = (id− tS)−1S

is diagonalizable as well.If N = Rξ is the affine normal bundle for a Blaschke hypersurface x , then,in general, N is not the affine normal bundle for xt . It is the affine normalbundle, if det Lt = det (id− tS) is a non-zero constant on M.


S diagonalizable

S – diagonalizable with eigenfunctions k1, ..., knHl ( l = 1, ..., n) – the normed elementary symmetric functions

Hl =

(nl

)−1 ∑1¬i1<...<il¬n

ki1 · ... · kil

and set H0 = 1.Assume that xt is an immersion. The shape operator St for xt isdiagonalizable. Denote by k1(t), ..., kn(t) the eigenfunctions of St . Then

St = L−1t S , ki (t) =ki1− tki

and consequently

ki =ki (t)1+ tki (t)

.

Let Hl(t), l = 1, ..., n, denote the normed elementary functions for xt .Barbara Opozda () On parallel submanifolds Leuven, August 2012 22 / 42

We shall say that a hypersurface x with equiaffine transversal vector field ξis H-linear Weingarten if there are real numbers a0, ..., an, not all zero,such that

Hnan + ...+ H1a1 + a0 = 0 (11)

at each point of M. The polynomial

W (t) = a0tn + ...+ an−1t + an (12)

will be called the associated polynomial for the H-linear Weingartenhypersurface.


A hypersurface with diagonalizable shape operator is Weingarten if there issome functional relationship between their principal curvature functionsk1, ..., kn. For instance, we can say that a hypersurface is k-linearWeingarten if there are real numbers c0, ..., cn, not all zeros, such that

kncn + ...+ k1c1 + c0 = 0

at each point of M. We can also consider the principal radii of curvaturefunctions and the corresponding normed elementary symmetric functionsand their linear relations.


The basic theorem is the following

Theorem

Let x : M → Rn+1 be an H-linear Weingarten hypersurface with equiaffinetransversal vector field, diagonalizable shape operator and satisfying

Hnan + ...+ H1a1 + a0 = 0.

Assume that xt is an immersion. Then xt is also H-linear Weingarten. Itsatisfies the condition

Hn(t)W (t) +11!Hn−1(t)W ′(t) +

12!Hn−2(t)W ′′(t) + ...+

+1

(n − 1)!H1(t)W (n−1)(t) +

1n!H0(t)W (n)(t) = 0,

where W (t) = a0tn + ...+ an−1t + an is the associated polynomial for x .

Note that 1n!H0(t)W(n)(t) = ao .


Theorem

Let x : M → Rn+1 be a hypersurface with an equiaffine transversal vectorfield, diagonalizable shape operator whose eigenvalues are mutually distinctat each point of M. If x is H-linear Weingarten satisfying the equation

Hnan + ...+ H1a1 + a0 = 0,

a0 6= 0 and the polynomial

W (t) = a0tn + ...+ an−1t + an

has a root t0 of multipicity (n − 1) but not of multiplicity n, then xt0 is animmersion with constant non-zero mean curvature H1(t0).


Proof. It follows from the previous theorem if we prove that xt0 is animmersion. Suppose it is not an immersion. Then0 = det Lt0 = (1− t0k1) · ... · (1− t0kn) at some point p of M. We canassume that t0 = 1

k1(p). Since a0 6= 0 we can assume that a0 = 1.The

polynomial associated with x is now of the form(t − 1

k1(p)

)n−1(t − a),

where a 6= 1k1(p). A contradiction follows now from

LemmaLet a, k1, ..., kn be real numbers such that k1 6= 0. If we have the equalityof polynomials

tn + a1tn−1 + ...+ an =(1− 1k1

)n−1(t − a), (13)

then

nkn−11 (1+ a1H1 + ...+ anHn) = (1− ak1)(k1 − k2)...(k1 − kn), (14)

where Hl =(nl

)−1∑1¬k1<...<il¬n ki1 · · · kil .


In a similar one can prove

Theorem

Let x : M → Rn+1 be a hypersurface with an equiaffine transversal vectorfield, with diagonalizable shape operator whose all eigenfunctions aremutually distinct at each point of M. If x is H-linear Weingarten and itsassociated polynomial is of degree (n − 1) and has a root t0 of multiplicity(n − 1), then xt0 is an immersion of vanishing H1(t0) (xt0 is minimal).


The case n = 2, S - not necessarily diagonalizable

x : M → R3 - immersion with an equiaffine transversal vector field ξ

H1 = 12trS , H2 = detS

PropositionLet x be an H-linear Weingarten surface satisfying the equality

a2H2 + a1H1 + a0 = 0. (15)

If xt is an immersion, that is,

0 6= det Lt = t2H2 − 2tH1 + 1, (16)

then xt is also H-linear Weingarten and he following equality is satisfied

0 = a0 + [2a0t + a1]H1(t) + [a0t2 + a1t + a2]H2(t). (17)


– the associated polynomial

W (t) = a0t2 + a1t + a2

– if a0 6= 0, it is a quadratic trinomial with ∆ = a21 − 4a0a2

Proposition

Assume that x : M → R3 is an H-linear Weingarten surface andH21 − H2 6= 0 at each point of M. If in its associated polynomial a0 6= 0,∆ 6= 0 and t1, t2 are the roots of the assiciated polynomial, then xt1 , xt2are immersions of constant mean curvature equal to H1(t1) = 1

t2−t1 ,H1(t2) = 1

t1−t2 .


Proposition

Assume that x : M → R3 is an H-linear Weingarten surface whose shapeoperator S is non-singular at each point of M. If for the associatedpolynomial a0 6= 0 and ∆ 6= 0, then there is t ∈ R such that xt is animmersion with constant curvature H2(t) =

4a20∆ .


Proposition

(a) If an immersion x : M → R3 with an equiaffine transversal vector fieldhas positive constant Gauss curvature H2 and H21 − H2 6= 0 at each pointof M, then there is t such that xt is an immersion of constant meancurvature H1(t) = 2√

H2.

(b) If the curvature H2 6= 0 at each point of M and H1 is non-zeroconstant, then there is t such that xt is an immersion of positive constantcurvature H2(t) = 4H21 .


Proof. (a)We can set a2 = 1H2, a0 = −1, a1 = 0. Since H2 > 0, we have

∆ > 0 and we can use the previous proposition. (b)We now set a2 = 0,a1 = −1, a0 = H1. Take t = 1

2H1. We now have

det Lt = 1− 212H1H1 +

14H21H2 =

14H2H216= 0,

which implies that xt is an immersion on the whole M. From (17) weobtain

0 = H1 + [H114H21− 12H1]H2(t) = H1 −

14H1H2(t).

Thus H2(t) = 4H21 .


Proposition

Assume that x : M → R3 is an H-linear Weingarten surface with anequiaffine transversal vector field whose shape operator S is non-singularand H21 − H2 6= 0 at each point of M. If in the associated polynomiala0 = 0, then there is t ∈ R such that xt is an immersion and H1(t) = 0 onM.

Proposition

Let x : M → R3 be an H-linear Weingarten surface with an equiaffinetransversal vector field for which H2 ¬ 0 on M. If a0 = 0 and a1 6= 0, thenthere is t ∈ R such that xt is an immersion and H1(t) = 0 on M.


One-parameter deformation of Blaschke hypersurfaces

x : M → Rn+1 – non-degenerate with the Blaschke structureξ – the affine normal vector field, N = R ξ – the affine normal bundlext = x + tξ – the one-parameter deformationLt = id − tS – the corresponding deformation tensorsWe already know that N is the affine normal bundle for xt if and only ifc(t) = det Lt is constant on M. If c(t) is this constant then the bundle Nis the affine normal bundle for xt but ξ is not necessarily the affine normalfor xt . The affine normal ξt for xt is equal to ξt = Φ(t)ξ where

Φ(t) = ε|c(t)|−1n+2

and ε is the sign of c(t). Denote by St the affine shape operator for xt .Then

St = Φ(t)St = Φ(t)L−1t S .


Assume that S is diagonalizable and xt is an immersion. Then St isdiagonalizable as well. If x is locally strongly convex, then xt does not haveto be locally strongly convex, but the shape operator for xt isdiagonalizable.As before we shall denote by kl , Hl , l = 1, ..., n, the affine principalcurvatures and the elementary symmetric functions determined by theaffine shape operator for x . The eigenfunctions of St , for xt are equal toΦ(t)kl(t). Hence the affine H-curvatures for xt are given by

Hk(t) = Φ(t)kHk(t).

The last formula is also valied if S is not diagonalizable and Hl are suitablynormed coefficients of the characteristic polynomial of the affine shapeoperators.


Proposition(a) If there is t0 such that xt0 is an immersion and N is the affine normalbundle for xt0 , then x is H-linear Weingarten. The H-curvatures satisfy theequality

0 = (1−c(t0))+(−t0)(n1

)H1+ ...+(−t0)l

(nl

)Hi + ...+(−t0)nHn. (18)

(b) The immersion xt0 is also H-linear Weingarten and H-linearWeingarten.(c) If n = 2, then

t20H2 − 2t0H1 + (1− c(t0)) = 0.

t20H2(t0) + 2t0H1(t0) +c(t0)− 1c(t0)

= 0,

t20 (c(t0))32 H2(t) + 2ε(c(t0))

54 H1(t0) + c(t0)− 1 = 0.


In particular, if c(t0) = 1, then

H2(t0)H1 = −H1(t0)H2.

If S is diagonalizable and non-singular at every point of M, then

P1 = −P1(t0),

where P1 = 12(1k1+ 1k2 ) and k1, k2 are the eigenvalues of S , and P is

defined in an analogous way.

In general, the H-curvatures satisfying (18) are not constant. In such acase, if n = 2, t0 is the only parameter (except for t = 0) for which N isits affine normal bundle. Namely, if xt is an immersion and N is its affinenormal bundle, then t2H2 − 2tH1 and t20H2 − 2t0H1 are constants. Hence2dH1 = tdH2 and 2dH1 = t0dH2. It is not possible unless H1 and H2 areconstant.


Assume now that all curvature functions H1,..., Hn are constant. Then c(t)is constant for all t. Hence if only c(t) is not zero, then N is the affinenormal bundle for xt . If c(t) = 0, then xt is not an immersion at any point.If S is diagonalizable, then using induction from n to 1 and the formula(

nk

)Hk(t) =

1c(t)

n−k∑j=0

(nn − j

)(−t)k−jHn−j

we see that all curvatures H1(t),,...,Hn(t) are constant. It follows thatH1(t), ..., Hn(t) are constant as well.


Using considerations from the section on surfaces equipped with equiaffinetransversal bundles we can prove

Proposition

Let x : M → R3 be a Blaschke surface with constant curvatures H1,H2and H2 < 0. There is t such that xt is a minimal Blaschke surface.


One can also consider the induced connections for xt . For example, wehave

PropositionLet x be a non-degenerate immersion with an equiaffine transversal vectorfield such that rkS > 1 everywhere on M. If for some t the mapping xt isan immersion such that R(t) = R, then S is a non-zero multiple of theidentity.


Proof. If connections ∇ and ∇ are related by ∇ = L−1∇L, then theircurvature tensors are related by

L R(u, v) = R(u, v) L.

From this formula and the assumption about the curvature tensors we getR · S = 0, that is, by the Gauss and Ricci equations we have

h(v ,Sw)Su − h(Su,w)Sv = h(v ,w)S2u − h(u,w)S2v (19)

for all u, v ,w tangent to M.Assume that there is a point p ∈ M such that S is not proportional to theidentity on the tangent space TpM. Take u ∈ TpM such that u,Su arelinearly independent. The spaces u⊥ (orthogonality relative to thenon-degenerate h), Su⊥ are (n − 1)-dimensional. They are not equal,because u and Su are linearly independent. Hence u⊥ \ Su⊥ 6= ∅.Take w from this set. By (19) we see that for each v ∈ TpM we haveSv ∈ R(Su), which contradicts the assumption rkS > 1.


On parallel submanifolds · Parallel totally real submanifolds RN = Cn – the complex aﬃne space...

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