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November 10, 2018
Abstract

In this paper we consider centroaffine codimension 2 immersions with an eqüiaffine transversal vector field. We prove that asymptotic lines, resp. inflection points, of an immersion correspond to principal lines, resp. umbilical points, of its dual. As a corollary, we obtain that the umbilical centroaffine immersions are given by affine cylindrical pedals. For curves in -space, we give a new proof of an Arnold’s four planar points theorem. For surfaces in -space, we prove a weak version of Loewner’s conjecture for asymptotic lines at inflection points. This last result is obtained by duality from a version of the Loewner conjecture for curvature lines at affine umbilical points of surfaces in -space.

Centro-affine Codimension 2 Immersions] Centro-affine Codimension 2 Immersions:
Umbilical and Inflection Points

M.Craizer]Marcos Craizer

R.A.Garcia]Ronaldo Garcia

The authors want to thank CNPq and CAPES for financial support during the preparation of this manuscript.
E-mail of the corresponding author: craizer@puc-rio.br.\par\@mkboth\shortauthors\shorttitle \par [\par

Mathematics Subject Classification (2010).  53A15, 53A04, 53A05, 37C10.

Keywords. Affine Umbilical Points, Affine Curvature Lines, Asymptotic Lines, Inflection Points, Loewner’s conjecture, Carathéodory conjecture..

\par\par\@xsect\par

We consider in this paper centroaffine codimension immersions, i.e., immersions such that the radial vector field does not belong to the tangent space , for any . We choose then a vector field along such that does not belong to , where we are identifying with and denotes the space generated by . \parFor and vector fields tangent to , write

(1.1)

where and are bilinear forms on and is a torsion-free connection on . The property of being non-degenerate does not depend on the choice of ([5]), and we shall assume it throughout the paper. Geometrically this condition means that if we locally project radially to an affine hyperplane of , we obtain a non-degenerate hypersurface, i.e., a hypersurface with non-degenerate Euclidean second fundamental form. \parWe can write

(1.2)

where and are -forms and is a linear map of . is called centroaffine shape operator and its eigenvectors are called principal directions. A curvature line is a line tangent to a principal direction at each point and an umbilical point is a point where is a multiple of the identity. \parWe say that the pair is eqüiaffine if . One can define the notion of dual immersion (see [5]) and verify that the dual of a non-degenerate eqüiaffine pair is another non-degenerate eqüiaffine pair . We prove in this paper that curvature lines, resp. umbilical points, of a pair correspond to asymptotic lines, resp. inflection points, of its dual. \parAny non-degenerate hypersurface with an eqüiaffine transversal vector field can be lifted to a non-degenerate centroaffine immersion together with an eqüiaffine vector field . The dual of the pair is called the affine cylindrical pedal of . We characterize the non-degenerate umbilical centroaffine immersions as affine cylindrical pedals of some non-degenerate pair . \parFor curves in -space, we give a new proof of an Arnold’s four planar points theorem ([1]). Our proof is obtained by duality from a four vertices theorem for planar convex curves with an eqüiaffine transversal vector field. \parFor surfaces in -space, we show a weak version of Loewner’s conjecture, which says that the index of an asymptotic foliation around an inflection point is . In order to prove it, we first prove a weak version of Loewner’s conjecture for curvature lines of an eqüiaffine pair at an elliptic umbilical point, where is a non-degenerate immersion and is an eqüiaffine transversal vector field. We discuss also the implications of these results to Carathéodory’s conjectures. \parThe paper is organized as follows: In section 2 we describe the known theory of codimension immersions. In sections 3, in case , and 4, in case , we prove the relation between curvature lines, resp. umbilical points, of an immersion and the asymptotic directions, resp. inflection points, of the dual immersion. In section 5 we use this relation to characterize the umbilical immersions as affine cylindrical pedals. In section 6, we give another proof of an Arnold’s four planar points theorem for spatial curves. In section 7 we prove a weak version of Loewner’s conjecture for curvature lines at an umbilical point of a surface in -space with an eqüiaffine transversal vector field and the corresponding Loewner’s conjecture for the index of asymptotic lines at an inflection point. \par\par\par\par\par\par\@xsect \par\par\parDenote by the flat affine connection of , by the radial vector field on and by a parallel volume form of . A centro-affine codimension 2 immersion is a map transversal to the radial vector field . In order to keep notations shorter, we shall use instead of . \par\par\parIn this section we follow [4, N9] and [5] and assume . \par\par\@xsect \par\parWe call a vector field along transversal if does not belong to , . For a transversal vector field , consider Equations (1.1) and (1.2). It is proved in [5] that the conformal class of is independent of the choice of . We say that the immersion is non-degenerate if the class of is non-degenerate. \par\par\parWe may write

where is the -form defined by

We say that is eqüiaffine if , which is equivalent to being parallel along . Along this paper, we shall assume that the transversal vector field is eqüiaffine. \par\parLet be a -orthonormal basis of . We say that a transversal eqüiaffine vector field is Blaschke if

(2.1)

An eqüiaffine vector field satisfying condition (2.1) is unique up to , i.e., the difference of two such vector fields is a multiple of ([5]). \par\par\par\par\@xsect \parAssume that is a non-degenerate immersion and an eqüiaffine transversal vector field. The dual of is defined by the following equations:

(2.2)
(2.3)
\par

The following lemma is proved in [5]: \par

Proposition 2.1.

Consider a non-degenerate centroaffine immersion and let be an eqüiaffine transversal vector field. Then is a non-degenerate centroaffine immersion and is an eqüiaffine vector field. Moreover, the dual of is and we can write

(2.4)
(2.5)

where

(2.6)

and

(2.7)
\par\par

Concerning Blaschke vector fields, we have the following proposition: \par

Proposition 2.2.

Consider dual eqüiaffine pairs and . Then is Blaschke if and only if is Blaschke.

\par

Before proving this proposition, we prove the following lemma: \par

Lemma 2.3.

We have that

Proof.

Differentiating the equation we obtain

which implies that . ∎

\par
Proof.

(of Proposition \@setrefprop:Blaschke). Consider a -orthonormal basis for and assume that is Blaschke. Then

By Lemma \@setreflemma:H, the basis is dual to , which implies that

where denotes the dual volume form of . Thus is also Blaschke. ∎ \par\par\par\par\par\par\par\@xsect \par\par\par\@xsect \parFor eqüiaffine immersions, the shape operator is self-adjoint with respect to the metric ([5]). If is positive definite, has eigenvalues, not necessarily distinct. Points with equal eigenvalues are called umbilical and, outside umbilical points, the eigenvectors of determine the principal directions. \parAt an umbilical point, write . Then and , for any tangent to (see Lemma 4.2 and Theorem 4.3 of [5]). We conclude that if all points are umbilical, then all planes generated by and contain a fixed line through the origin. \par\par\par\par\@xsect \parThe second fundamental forms with respect to and are exactly and . Thus, in case , the following definition agrees with [8, p.214]: \par

Definition 3.1.

We say that is an asymptotic direction at if there exists such that . If , then we say that is an inflection point.

\par
Lemma 3.2.

We have that is an asymptotic direction for at if and only if there exists such that

(3.1)

for any .

Proof.

Equation (3.1) is equivalent to

which proves the result. ∎

\par

The following proposition is proved in [4, Prop.N9.3] \par

Proposition 3.3.

The image of the immersion is contained in an affine hyperplane if and only if , for some scalar function .

\par\par\par\par\par\par\@xsect\par

Consider a dual pair and of non-degenerate eqüiaffine pairs. \parWe shall verify below that umbilical points of correspond to inflection points of . Moreover, the principal directions of correspond to the asymptotic directions of . \par\par\par\par

Proposition 3.4.

A principal direction for corresponds to an asymptotic direction for .

Proof.

We have that is an asymptotic direction for if

for any , which is equivalent to

Thus is an asymptotic direction for if and only if

for any . Since is non-degenerate, this is equivalent to , which is the condition for to be a principal direction for . ∎

\par\par\@xsect\par

In this section we describe the concepts of sections \@setrefsec:2 and \@setrefsec:3 in the case . \par\par\@xsect \parConsider a curve such that and are linearly independent. Let be a vector field along such that

We can write

(4.1)

where , and are scalars functions on . The non-degenerate condition can be also written as

(4.2)

which is equivalent to say that the osculating plane is well defined and the vector field is transversal to it. A curve is centroaffine non-degenerate if and only if its local radial projection with respect to the origin in a plane is locally convex. \parFor an eqüiaffine vector field , we write

(4.3)

where and are scalars functions on . \par\par\@xsect \parWe say that is a vertex of the eqüiaffine pair if =0. Observe that this condition implies that . Thus if all points of a curve are vertices, then all planes generated by and contain a fixed line through the origin. \par\parWe say that the point is a planar point for the immersion if belongs to the osculating plane of the curve at . Take . Then belongs to the osculating plane, i.e.,

(4.4)
\par
Lemma 4.1.

The point is planar if and only if .

Proof.

Differentiating Equation (4.4) we obtain that is in the osculating plane if and only if

which is equivalent to . ∎

\par
Proposition 4.2.

The curve is contained in a plane if and only if , .

Proof.

Under the non-degenerate hypothesis (4.2), the condition belongs to the osculating plane, for any , is equivalent to the curve being contained in a plane. ∎

\par\par\par\par\par\par
Proposition 4.3.

Let and be dual non-degenerate eqüiaffine pairs. Then

(4.5)

and

(4.6)
\par
Proof.

Since is eqüiaffine and , we can write

for certain scalar function . Write

(4.7)

where . Now Equation (4.4) can be written as

Thus

and we conclude that , thus proving formula (4.6). Writing

we obtain and . Finally

thus proving formula (4.5). ∎

\par\par\par\par\par\par
Proposition 4.4.

A vertex of corresponds to a planar point of .

Proof.

It follows from Equation (4.6) that a point is a vertex of if and only if . On the other hand, it follows from Lemma \@setreflemma:Planar that is planar for if and only if . ∎ \par\par\@xsect \parConsider the centroaffine arc-length parameter for , i.e., reparameterize such that

(4.8)

and take . Then is the affine normal vector field of the curve and we can write

(4.9)

From Equation (4.4), we obtain . The dual centroaffine immersion is given by

(4.10)

Observe that and so is the affine normal vector field of the curve . Moreover,

(4.11)

A point is planar for and a vertex for if and only if . \par\par\par\par\par\par\par\par\par\par\@xsect \parThe results of this section are valid for . \par\par\@xsect \parConsider a non-degenerate immersion with an eqüiaffine transversal vector field . Let be given by

(5.1)

Then is a centroaffine immersion and are linearly independent. We call the pair the lifting of . \parDenote by and the metric and the shape operator of , respectively. It is easy to verify that , where denotes the metric of , which implies that is a non-degenerate immersion. One can also verify that is eqüiaffine and that , where denotes the shape operator of . Since is contained in a hyperplane, for any , i.e., all points are inflection points. \parConversely, if all points of an immersion are inflections, then is contained in a hyperplane. Thus we may assume that and , where is a centroaffine immersion and is eqüiaffine. \par\par\par\par\@xsect \parDenote by the co-normal map associated to , i.e.,

The affine cylindrical pedal is defined as

(5.2)
\par
Lemma 5.1.

Let . Then is transversal to and the pair is dual to .

Proof.

Straightforward verifications. ∎

\par
Proposition 5.2.

Consider an umbilical eqüiaffine immersion . Then necessarily is an affine cylindrical pedal and is a constant vector field.

\par
Proof.

If is umbilical, then its dual is hyperplanar. By Section \@setrefsec:Hyperplanar, is the lifting of some immersion with eqüiaffine transversal vector field , which proves the proposition. ∎ \parWhen is a Blaschke vector field for , we have the following corollary: \par

Corollary 5.3.

Consider an umbilical non-degenerate Blaschke immersion . Then is an affine cylindrical pedal of a Blaschke immersion and is a constant vector field.

\par

For a similar result in a slightly different context, see [7]. \par\par\@xsect \parNext proposition says when a non-degenerate immersion can be considered as an affine cylindrical pedal. \par

Proposition 5.4.

Consider a non-degenerate centroaffine immersion transversal to . Then is the affine cylindrical pedal of an eqüiaffine non-degenerate immersion .

Proof.

Since is transversal, is a non-degenerate eqüiaffine umbilical pair. By Proposition \@setrefprop:UmbilicalACP, is an affine cylindrical pedal. ∎ \par\par

Corollary 5.5.

Consider a non-degenerate centroaffine immersion . Then is locally the affine cylindrical pedal of an eqüiaffine non-degenerate immersion .

\par
Proof.

We may assume that locally the radial projection in the plane is a non-degenerate hypersurface and that does not belong to it. This implies that is transversal to , which proves the corollary. ∎

\par\par\par\@xsect\par

In this section we recall a four vertices theorem for planar convex curves together with an eqüiaffine transversal vector field. We show that the dual result is a four planar points theorem for spatial curves whose radial projection in a plane is convex, thus providing another proof of a well-known Arnold’s theorem ([1]). \par\par\par\@xsect \parWe say that a closed planar curve is convex if any line touches it in at most points. It is well-known that a locally convex planar curve is convex if and only if its index is . \parConsider a closed convex planar curve and an eqüiaffine vector field . We write

(6.1)

A point is a vertex of if . In this context, the following theorem holds: \par

Theorem 6.1.

The eqüiaffine pair has at least four vertices.

\par

This vertices theorem is a consequence of the vertices theorem for exact transversal vector fields along a planar convex curve (see [10]) and the fact that any eqüiaffine transversal vector field is exact ([9, Th.2.8]). \par\par\par\par\par\par\par\@xsect \parRecall that the affine cylindrical pedal of a closed locally convex planar curve together with an eqüiaffine vector field is given by

where is the co-normal vector field of associated with . \par

Lemma 6.2.

Assume that , with and convex. Then the planar curve is convex.

Proof.

Since is locally convex, we have only to show that its index is . But the index of is the same as the index of its co-normal vector field . On the other hand, since is convex, the curve intersects the rays from the origin at most once. Thus its index is , which completes the proof of the lemma (see Figure \@setreffig:Fig1). ∎ \par\par

Figure 1: A spatial curve transversal to and its dual pair . Note that the radial projection of is convex, while is only starred.
\par\par\par

We now give a new proof of the following theorem ([1]): \par

Proposition 6.3.

Assume that is a closed curve in whose radial projection in an affine plane is convex. Then has at least four planar points.

\par
Proof.

Write , with is convex. We may assume that is contained in the region bounded by , which implies that is transversal to . \parBy Proposition \@setrefProp:CharPedal, is the affine cylindrical pedal of a planar eqüiaffine pair . By Lemma \@setreflemma:Convex, is convex. Since by Theorem \@setrefthm:4Vertices has at least four vertices, the curve has at least 4 planar points. ∎ \parFor other planar points theorems of space curves, see [11]. \par\par\par\@xsect \par\par\parConsider an immersion . Denote by an eqüiaffine transversal vector field and by the corresponding shape operator, i.e.,

An umbilic point is a point such that is a multiple of the identity. The shape operator is self-adjoint with respect to the affine metric . Assuming is positive definite, has exactly eigenvectors at each non-umbilic point. The directions of the eigenvectors are called principal and their integral lines are called curvature lines. \par\par\parIn the Euclidean case, the Carathéodory conjecture states that any compact surface homeomorphic to the sphere admits at least Euclidean umbilic points. This conjecture is a consequence of Loewner’s conjecture, which states that the index of the Euclidean curvature lines at an umbilic point is at most . For the controversies concerning these conjectures, see [6]. Loewner’s conjecture can be also be stated for the asymptotic foliations of a -dimensional immersion in -space at an inflection point (see [2]). \par\par\@xsect \parIn this section we state a weak version of Loewner’s conjecture for eqüiaffine immersions. Let

(7.1)

where is the shape operator. Observe that a point is an isolated affine umbilical point for if and only if it is an isolated zero of . We say that an isolated umbilical point is semi-homogeneous if it is also an isolated zero of the first non-zero jet of at . The weak version of Loewner’s conjecture is the following: \par\par

Theorem 7.1.

Assume is a semi-homogeneous affine umbilical point of an eqüiaffine immersion . Then the index of the curvature line foliation is at most .

\par

Consider a centroaffine non-degenerate immersion and let be an inflection point. By Proposition \@setrefProp:CharPedal, is the affine cylindrical pedal of some eqüiaffine pair , and by Proposition \@setrefProp:PrincipalAsymptotic, is an umbilical point for this pair. We say that is a semi-homogeneous inflection point for if it is a semi-homogeneous umbilical point for . \par\par

Theorem 7.2.

Assume is a semi-homogeneous inflection point of an immersion . Then the index of the asymptotic line foliation is at most .

\par
Proof.

By Proposition \@setrefProp:CharPedal, is the affine cylindrical pedal of some eqüiaffine pair , and by Proposition \@setrefProp:PrincipalAsymptotic, the asymptotic foliations of correspond to the principal foliations of . Then Theorem \@setrefthm:Loewner1 implies the result. ∎ \parIn view of Theorem \@setrefthm:Loewner1, it is natural to propose the following conjecture: \par\par\@xsect Assume is an affine umbilical point of an eqüiaffine immersion . Then the index of the curvature line foliation is at most . \parFrom Propositions \@setrefProp:CharPedal and \@setrefProp:PrincipalAsymptotic, we have an equivalent conjecture for asymptotic lines of an immersion of a surface in -space: \par\par\@xsect Assume is an inflection point of an immersion . Then the index of the asymptotic line foliation is at most . \parIt is well-known that Conjecture 2 holds generically. In fact, it is proved in [3] that for a generic immersion , the index of an inflection point is at most . \par\par\par\par\@xsect \par\parWe say that are isothermal coordinates on if

It is well-known that any convex surface can be covered by isothermal parameterizations. \parOne can verify that the shape operator is given by

or equivalently,

(see [4, N4, p.208]). \parConsider the affine support function defined by

(7.2)

where is the co-normal vector field associated to and is a fixed point. Denote the affine support function in isothermal coordinates. Observe that

Differentiating we obtain

It follows that

(7.3)

Denote

(7.4)
Lemma 7.3.

Assume . Then the point is umbilical if and only if it is a zero of .

Proof.

If , then Equation (7.3) implies that

Thus if and only if . ∎

\par\par\par\par\@xsect\par\par\par
Lemma 7.4.

Assume that is umbilic and let , . Then

(7.5)

Conversely, if equation (7.5) is satisfied, then is umbilic and .

\par
Proof.

The equation holds up to order if and only if

at , which is equivalent to say that is umbilic. ∎

\par\par\par\par\par
Proposition 7.5.

Let be umbilic and . Then

(7.6)

where and denote the Jacobian matrix of the vector fields and , respectively.

\par
Proof.

We shall verify the equality of the derivatives of and with respect to , which corresponds to the -entries of both matrices, the other entries being similar. Observe that

At , and so

On the other hand, since , we have that

Thus, at ,

thus proving the desired result. ∎

\par

It follows from Lemma \@setrefLemma:UmbilicalP and Proposition \@setrefProp:jet1P that, at simple umbilical points and up to order , we can replace by . \par\par\par\par\@xsect \parWe say that is umbilic of order if the -jet of vanishes. \par

Lemma 7.6.

Let be umbilic and . Then is umbilical of order if and only if

(7.7)
\par
Proof.

Write

(7.8)
\par

If the -jet of at equals , differentiating equation (7.8) times we obtain that, at , the -jet of equals times the -jet of , thus proving formula (7.7). \par\parAssume that equation (7.7) holds with . Differentiating the Equation (7.8) and taking , we obtain

which implies that . Similarly, we can prove that , which implies that the -jet of vanishes at . To prove that the -jet of vanishes at , one can proceed by an easy induction. ∎

\par\par
Proposition 7.7.

At an umbilic point of order ,

(7.9)

where and denote the -jets of and , respectively.

\par
Proof.

For , we must prove that the second derivatives of and are multiples at . Let us consider and , the other cases being similar. Observe that

At ,

On the other hand,

At ,

thus proving the proposition. To prove Equation (7.9) for any , one can proceed by induction. ∎

\par\par\@xsect\par\par\par\par
Lemma 7.8.

If at is semi-homogeneous of degree , the index of at is the same index of at .

\par
Proof.

For sufficiently small,

at , which proves the lemma. ∎

\par

For an isolated umbilic point , denote by the -jet of . \par

Corollary 7.9.

If at is semi-homogeneous of degree , the index of at is the same index of at .

\par

We can now prove theorem \@setrefthm:Loewner1. \par

Proof.

It is proved in [6] that any vector field of the form has index . In fact, since is homogeneous, this is the easy part of the proof given in [6]. ∎

\par\par\@xsect\par

As in the Euclidean case, Loewner’s conjecture implies Carathéodory’s conjecture. For example, Theorem \@setrefthm:Loewner1 and \@setrefthm:Loewner2 imply the following corollaries: \par

Corollary 7.10.

Consider a non-degenerate centroaffine immersion , compact, and an eqüiaffine normal vector field . Assume that all umbilical points are semi-homogeneous. Then there are at least umbilical points.

\par
Corollary 7.11.

Consider a non-degenerate centroaffine immersion , compact. Assume that all inflection points are semi-homogeneous. Then there are at least inflection points.

\par

On the other hand, if Conjecture 1 (or Conjecture 2) of Section \@setrefsec:Conjectures holds, then the following conjectures also hold: \par\par\@xsect Consider a non-degenerate centroaffine immersion , compact, and an eqüiaffine normal vector field . Then there are at least umbilical points. \par\par\@xsect Consider a non-degenerate centroaffine immersion , compact. Then there are at least inflection points. \par\par\par\@xsect \parAre there compact surfaces in -space with only two affine umbilical points? A rotational ellipsoid has exactly two Euclidean umbilical points. It is thus natural to look for compact rotational surfaces with only two affine umbilical points. The surprising fact is that if we consider rotational surfaces with the Blaschke transversal vector field, then there exists at least one umbilical parallel. \par\par\@xsect \parConsider a surface graph of a function of the form

Then, by an affine change of variables, the -jet of at is given by

Since

we obtain and , which implies that is affine umbilic. We conclude that the axes points of a rotation surface are affine umbilic. \par\par\@xsect \parAssume is a smooth rotation surface generated by the convex planar arc , , such that and , for . A parameterization of is given by

(7.10)

Then

Thus

Moreover,