versión On-line ISSN 1669-9637
Rev. Unión Mat. Argent. v.47 n.1 Bahía Blanca ene./jun. 2006
Partially supported by FAPESP and CNPq.
Abstract: This is a short, elementary survey article about taut submanifolds. In order to simplify the exposition, we restrict to the case of compact smooth submanifolds of Euclidean or spherical spaces. Some new, partial results concerning taut 4-manifolds are discussed at the end of the text.
This is a short, elementary survey article about taut submanifolds. In order to simplify the exposition, we restrict to the case of compact smooth submanifolds of Euclidean or spherical spaces. Sections 2 through 4 collect basic definitions and general results. In Section 5, we explain in greater detail some useful techniques that we use in Section 6 to prove some new, partial classification results about taut -manifolds.
Before introducing the concept of tautness, it is instructive to discuss the STPP. Let be a compact surface embedded in an Euclidean sphere . We say has the spherical two-piece property, or STPP for short, if is connected whenever is a closed ball in [Ban70]. The STPP is equivalent to requiring that every Morse distance function of the form
has exactly one local minimum. Intuitively, this can be seen by remarking that the sublevel sets of the are exactly the closed balls . Moreover, by replacing by , one sees that these conditions imply that every Morse distance function also has exactly one local maximum. Let be the th Betti number of where, unless explicitly stated, we will always use coefficients to simplify the exposition, and let be the number of critical points of a Morse distance function . From the Morse relations for the Euler characteristic,
one finally deduces that the STPP is equivalent to the fact that for all and for all such that is a Morse function.
Since the STPP is defined in terms of intersections with closed balls, it is immediate that it is a conformally invariant property of compact surfaces in . Of course, one can also consider the STPP for compact surfaces of Euclidean space. Since stereographic projection is a conformal transformation, the theories in and are equivalent.
It follows from results of Kuiper and Banchoff (see [Ban70]) that a compact surface substantially embedded in with the STPP is a round sphere or a cyclide of Dupin in , or the Veronese embedding of a projective plane in . Recall that an embedding into a sphere is called substantial if its image does not lie in a sphere of smaller dimension. Of course, it is enough for us to consider only substantial embeddings.
We finally note that the STPP condition is also equivalent to requiring that the induced homomorphism
in ech homology is injective for every closed ball . The use of ech homology allows one to use all closed balls rather than only those determined by level sets of distance functions that are Morse functions. If one prefers to use singular homology, than the equivalent condition is reformulated in terms that the above homomorphism in singular homology be injective for almost every closed ball.
Now we come to the main concept in this text. Let be a compact submanifold embedded in . We say is taut if the induced homomorphism
in ech homology is injective for every closed ball and for all . This is equivalent to requiring that every Morse distance function satisfies for all , i.e. is -perfect. Taut submanifolds were first considered by Carter and West [CW72]. It is clear that tautness is equivalent to the STPP in the case of surfaces. In the case of -manifolds, there is only a classification of the diffeomorphism types of manifolds admitting taut embeddings, which was given by Pinkall and Thorbergsson [PT89]. They remark that a geometrical classification, if possible, is expected to be a complicated problem since most of the examples already known depend on many parameters. Their result is that there are seven types of such -manifolds, namely , , , , , and . Here is the quaternion group, and is a diffeomorphism of that acts on each factor as the antipodal map. A classification of -manifolds admitting taut embeddings along similar lines seems feasible, and we will come back to this point in the last section.
Next we present other classes of examples of taut submanifolds. Clifford tori and the standard embeddings of projective spaces are important examples. We will not justify this assertion now, since these examples will be generalized below. It follows from the Chern-Lashof theorem [CL57] that the sphere can only be tautly embedded in Euclidean space as a round hypersphere in an affine subspace. This result also admits an alternative proof for by noting that the homology of is trivial except in dimensions and , which implies that a Morse distance function to a taut embedding of can have only critical points of index and , and so a focal point is necessarily of multiplicity , which yields that is umbilic according to an argument sketched below in Section 5 (compare [NR72]). Moving towards manifolds with more complicated topology, Cecil and Ryan proved that a taut -dimensional compact hypersurface of with the same homology as has precisely two principal curvatures at each point, and the principal curvatures are constant along the corresponding curvature distributions [CR78]. They called such a hypersurface a (high-dimensional) cyclide of Dupin. Thorbergsson considered the case of a taut compact submanifold of dimension that is -connected but not -connected, and showed that it must be either a cyclide of Dupin diffeomorphic to or the standard embedding of a projective plane over one of the four normed division algebras [Tho83b].
In another development, Bott and Samelson proved that the orbits of the isotropy representations of the symmetric spaces, sometimes called -representa-tions, are tautly embedded by explicitly constructing cycles forming a basis in -homology for these orbits [BS58]. These orbits are known as the generalized flag manifolds. The generalized flag manifolds are homogeneous examples of another very important, more general class of submanifolds, called isoparametric submanifolds. Hsiang, Palais and Terng proved that isoparametric submanifolds and their focal submanifolds are taut [HPT88].
Most of the known examples of taut embeddings are homogeneous spaces. On the other hand, Thorbergsson derived some necessary topological conditions for a homogeneous space to admit a taut embedding, which allowed him to exhibit some examples of homogeneous spaces which cannot be tautly embedded, e.g. the lens spaces distinct from the real projective space [Tho88].
Thorbergsson and I approached the problem of classifying taut compact submanifolds of Euclidean space which are extrinsically Riemannian homogeneous by first studying taut representations, namely those representations of compact Lie groups all of whose orbits are tautly embedded. The only examples known at that time were the -representations, and many proofs had been given of the tautness of special cases of generalized flag manifolds where the arguments were easier. In [GT03] (see also [GT00]), we classified the taut irreducible representations of the compact Lie groups. It turns out that the classification includes three families of representations that are not -representations, thereby supplying many new examples of tautly embedded homogeneous spaces. These families are given by the following table, where .
It is worth noting that these representations are exactly those representations of cohomogeneity three that are not -representations. In [GT02], we showed that the orbits of these representations also admit cycles of Bott-Samelson type.
The proof of the classification theorem in [GT03] is long and intrincate. It starts with a remark by Kuiper that implies that a taut irreducible representation has the property that the second osculating spaces of all of its nontrivial orbits coincide with the ambient space [Kui61]. We call representations with this property, irreducible or not, of class . The class is much more easy to deal with since it involves an infinitesimal condition. Indeed, we establish necessary upper bounds for the Dadok invariant (which is an integral algebraic invariant of an irreducible representation, see [Dad85]) for irreducible representations of class , which allows us to reduce a lot the size of that class in such a way that the remaining cases are treated with geometrical methods that we develop in the second part of the proof.
There are three main strategies involved in these methods. The first one often works a kind of induction. It states that any slice representation of a taut representations inherits the property of being taut. (Recall that the slice representation of a representation at a point is the representation induced by the isotropy subgroup at on the normal space to the orbit through at .) The second one might be called "the fundamental result about taut sums". It relates the topology of the orbits of a taut reducible representation to the topology of the orbits of its summands. Finally, the third strategy invokes a reduction principle in transformation groups which grosso modo reduces the task of deciding whether a given representation is taut or not to the study of the tautness of a much simpler representation with trivial principal isotropy subgroup.
The combination of the above three strategies is also effective in the classification of taut reducible representations of compact simple Lie groups which I completed in [Gor04]. In this case, since every orbit of a summand of a taut reducible representation is also an orbit of the sum, it follows from the classification in the irreducible case that every irreducible summand of a taut reducible representation is an -representation, and we need only to decide which sums of -representations remain taut. Each compact simple Lie group admits few -representations, so that the analysis via the geometrical methods discussed in the previous paragraph can be effectively carried out and the final result is given in the following table.
We make two simple remarks regarding the representations appearing in this table. The first one is that in the case of the first representation in the table, any number of summands in the sum can be replaced by the dual representation , and the resulting representation remains taut. The second one is that the representations of are listed up to composition with an outer automorphism of the Lie group, so the pair appearing in the list can be replaced by any pair of inequivalent -dimensional representations of , and the resulting representations for will still be taut.
Despite these results, the classification of taut extrinsically Riemannian homogeneous submanifolds of Euclidean space is still far from complete. For example, we still do not know what the taut reducible representations of the nonsimple groups are.
In this section, we review a collection of methods related to taut submanifolds that were used in [PT89] and will be useful here.
Let be a compact smooth -manifold substantially embedded in . Later we will specify to the case in which is tautly embedded. Denote by the unit normal bundle of in . A focal point of in is a critical value of the restriction of the exponential map of to . According to this definition, is a focal point of for every if . This kind of focal point is uninteresting for us since it does not relate to the geometry of , but rather to the fact that any pair of antipodal points are conjugate in the sphere. Therefore in the following we will completely disconsider this kind of focal point. All the other focal points of can be described as follows. For , the set of focal points of lying in the normal ray defined by is discrete and in correspondence with the principal curvatures of relative to the Weingarten operator in such a way that for a focal distance there corresponds an eigenvalue of .
Now it can be checked that a point is a focal point of if and only if the distance function is not a Morse function. In this case, is a focal point relative to some , is a degenerate critical point of , and the multiplicity of as a focal point relative to equals the nullity of the Hessian of at , which is also the the multiplicity of the corresponding principal curvature of . For a nonfocal point , is a Morse function and the Morse index theorem asserts that the index of a critical point of equals the sum of the multiplicities of the focal points of lying in the normal geodesic segment .
It follows that the sum of the multiplicities of the focal points lying in any open half great circle joining and is . Since and , we also have that for each focal distance in the direction of , there is a corresponding focal distance in the direction of with the same multiplicity. Let denote the number of distinct focal points lying in the open half great circle specified by . We have that , as a function on , is lower semicontinuous and there is an open and dense subset of where is locally constant and maximal. A normal vector is called regular if it belongs to , and singular otherwise.
We suppose henceforth that is tautly embedded in .
Now the condition that every unit normal vector is regular is equivalent to the condition that be a proper Dupin submanifold of [Pin86, Tho83a]. The latter means that the principal curvatures are constant along the corresponding curvature surfaces and the number of distinct principal curvatures is constant. In this case, there is a well defined ordered sequence of multiplicities of focal points of along the normal ray defined by an arbitrary . Here is the (constant) number of distinct principal curvatures of .
If is taut and there exists a unit normal vector such that , then is umbilic, and hence, a great hypersphere in . This can be seen as follows. Let be the first focal point of in the direction of . The multiplicity of is by hypothesis. Consider the geodesic segment , the open half great circle containing it, and a point belonging to . The Morse index theorem implies that the index of as a critical point of is or , according to whether occurs between and or past . It follows that is a local minimum or a local maximum of , respectively. By tautness of , there can be only one local minimum (resp. local maximum), so this is also a global minimum (resp. global maximum). Letting tend to in both cases shows that is contained in the hypersphere of center and radius , and hence coincides with that hypersphere.
As was said above, if is taut and is a focal point, then is not a Morse function, but there is a very useful theorem by Ozawa [Oza86] that asserts that is a -perfect Morse-Bott function and the connected components of the critical set of are compact smooth taut submanifolds of . Recall that a function is called a Morse-Bott function if the connected components of its critical set are smooth submanifolds along each of which the nullity of the Hessian of is constant and equal to the dimension of the component [Bot54]. In this case, the index of along a critical manifold is by definition the index of the Hessian of restricted to the normal bundle of that critical manifold. A Morse-Bott function on is perfect if it satisfies the Morse-Bott equalities, namely the Poincaré polynomials of and those of the critical manifolds of are related by the formula
If is taut, it follows from Ozawa's theorem and the Morse-Bott equalities that the -homology of the critical manifold corresponding to the set of minima (resp. maxima) of a distance function injects into the -homology of . Such critical sets are called top sets or top cycles, or yet top circles, top tori, etc., when their topology is specified, following terminology introduced by Kuiper and expanded by Pinkall and Thorbergsson. For each , we denote by the top cycle which is the set of minima of , where is the first focal point in the normal ray defined by .
Another interesting consequence of Ozawa's theorem is that the definition of a taut submanifold can be restated so as to require that the homomorphism (1) in singular homology be injective for all closed balls .
In this section we present some new, partial results about the classification of compact smooth manifolds of dimension four that admit taut embeddings. We start with the two following results, the first one of which has already been noticed by Thorbergsson.
Proof. If , then is a -homology sphere, and we have already seen that it must be a hypersphere in .
Next, we remark that a taut submanifold with must be simply connected. This is because it admits a -perfect Morse function, and such a function has no critical points of index one (compare Lemma 4.11(1) in [CE75]).
If , it follows from the preceeding remark that is a taut -manifold that is -connected but not -connected. Therefore Theorem A in [Tho83b] says that either is a cyclide of Dupin of type in , or is diffeomorphic to and sits inside . Recall that the standard embedding of in is taut.
Proof. By Theorem 1, we may assume that . Then, by hypothesis and -Poincaré duality, exactly four -homology groups of are nonzero. It follows from Theorem 3.2 in [Oza86] that (this also follows from Theorem 11 in [Heb88]).
We will use some basic facts about the topology of -manifolds [GS99]. Since there is a Morse function on with one critical point of each index , , , , there is a handle decomposition , where each an -handle attached via an attaching map.
If is orientable, then is an orientable handle, and are both diffeomorphic to , and is obtained by glueing two copies of along their boundaries by a diffeomorphism of . The diffeomorphism class of only depends on the diffeotopy class of . It has been shown that the diffeotopy group of is isomorphic to , where each generator extends to a diffeomorphism of [Glu62]. It follows easily that there is at most one manifold in this case, which plainly is .
If is nonorientable, then is a nonorientable handle, and are both diffeomorphic to , the twisted -disk bundle over , and is obtained by glueing two copies of along their boundaries via a diffeomorphism of . Every diffeomorphism of is diffeotopic to one that extends to [KR90]. It follows as above that must be diffeomorphic to , but the following lemma shows that this case cannot occur. This finishes the proof.
Proof. Suppose, on the contrary, that admits a taut and substantial embedding into a sphere . Since , any top set must have vanishing second Betti number. It follows that cannot admit a -dimensional top set, and any -dimensional top set has the -homology of . Therefore, any top set of , being taut, is diffeomorphic to or . This implies that there is no with . For otherwise, if is a point lying in between the second and third focal points in the normal ray defined by , and is the foot point of , then is a critical point of index of , but .
Now, for every , or . We claim that is proper Dupin. If not, there exist a singular with and a sequence converging to with . By replacing and the by their opposites if necessary, we may assume that the top set is diffeomorphic to . The are round circles representing nontrivial one-dimensional homology classes of . A subsequence of the converges to a circle in that is homologically nontrivial, and this is a contradiction (compare the proof of Theorem 2.2 in [Oza86]).
Now is proper Dupin. Suppose first that for all . We cannot have , since is not a cyclide of Dupin, so . Note that the multiplicities of the principal curvatures are and . Let be a small tube around . Then is a proper Dupin hypersurface with three distinct principal curvatures of multiplicities , and . It is known that a proper Dupin hypersurface with distinct principal curvatures has all multiplicities equal [Miy84], so this case cannot occur.
Suppose now that for all . A small tube around is a proper Dupin hypersurface with distinct principal curvatures of multiplicities . It is known that in the case of distinct principal curvatures there are at most two different multiplicities [GH91]. It follows that . Let and be the two components of and assume that . It follows from the Mayer-Vietoris sequence that
Remarks 1. (i) As a consequence of Theorem 2, a compact smooth -manifold with the same homology as which is tautly embedded in arbitrary codimension is diffeomorphic to (compare Theorem 2 in [CR78] and the next item (ii)).
(ii) According to Theorem 3.1 in [Oza86], the codimension of a taut and substantial embedding of into a sphere is either or . Both codimensions can be realized. In fact, can be realized as a cyclide of Dupin in codimension one, and it is shown in [CR85] that any taut embedding of with codimension one is such a cyclide, and they are all Möbius equivalent to a tube around a circle in . Moreover, can also be tautly embedded in as a singular orbit of the isotropy representation of the Grassmann manifold .
(iii) We have that , where acts by the antipodal map on each factor, is diffeomorphic to . In fact, and is a principal -bundle with a global section given by
(iv) According to Theorem 8 in [Heb88], cannot be tautly embedded into a sphere.
Regarding the following conjecture, note that the principal orbits of the isotropy representation of the Grassmann manifold are proper Dupin hypersurfaces in diffeomorphic to .
We next present a proof of this conjecture that uses that the Poincaré conjecture in dimension is true (there has been recent progress by Perelman on this conjecture, see [Mor05]). Let be a compact embedded proper Dupin hypersurface in with four distinct principal curvatures. Then the multiplicities of focal points along a normal ray are . For a fixed unit normal vector field, let (resp. ) denote the set of focal points corresponding to the largest (resp. smallest) principal curvature, and let be the focal maps. Then these are circle bundles whose fibers consist of the circles of curvature. It follows from the proof of Proposition 3.5 in [GH91] that and or and , and we may assume we are in the first case just by replacing the normal field by its opposite if necessary. Now the universal covering manifold of is a simply connected compact -manifold, hence diffeomorphic to if the Poincaré conjecture is true. The quotient of by a fixed point free involution is diffeomorphic to according to [Liv60], so is diffeomorphic to . So far we know that is a circle bundle over . Besides the trivial one, there are exactly two other circle bundles over , one oriented, one nonoriented (in fact, the associated -plane bundles are and , where is the trivial line bundle and is the tautological line bundle [Lev63]). Let , respectively denote the total spaces of these bundles. Using the Gysin sequence in the oriented case, one easily computes that , and hence, . Using the Serre homology spectral sequence in the nonoriented case, it is also not difficult to see that is a -extension of . In any case, and do not have the integral homology of the trivial circle bundle . But must have the integral homology of according to [GH91]. It follows that is diffeomorphic to .
It is easy to construct other examples of taut embeddings of compact smooth -manifolds that are not covered by the previous results, so there is still a lot of work to be done in order to achieve a complete classification. In the case of taut hypersurfaces in with no more than three distinct principal curvatures at each point, we have the following theorem [Nie91].
Theorem 3 (Niebergall). Let be a compact smooth taut hypersurface in . If has at most three distinct principal curvatures at each point, then is diffeomorphic to one of the following -manifolds: , , , , or the nontrivial -bundle over .
Niebergall proves this theorem by first remarking that there is an open and dense subset of where the multiplicities of the principal curvatures are locally constant. Then each connected component of is a piece of a proper Dupin hypersurface with three distinct principal curvatures in . By the main result of [Nie91], each connected component of is then a reducible proper Dupin hypersurface [Pin85], and hence can be reconstructed from lower dimensional proper Dupin hypersurfaces, where the possibilities for these are already known. The final step of the argument uses an unpublished result of Pinkall stating that a taut hypersurface is analytic.
We finish this text with the following lemma which refers to the case of a taut hypersurface in with generically four distinct principal curvatures.
Proof. Let with . Then the multiplicities of the focal points in the direction of are . Let be a point in the normal ray issuing from in the direction of that comes after the second focal point and before the third one. Then has a critical point of index at , so . Upon a choice of unit normal vector field on , there is an identification of with . Let be the restriction of the endpoint map. The focal set of consists of the critical values of and consists of focal varieties of codimension at least , hence its complement is connected. Let . Then is a local diffeomorphism. For , let be the index of as a critical point of for . Then is locally constant. Decompose into a disjoint union of nonempty open subsets
where . If , then is a diffeomorphism. This implies that is connected, but it is clear that must contain points on both sides of , so this is a contradiction. Hence, . Similarly, .
We would like to thank Professor D. L. Gonçalves and Professor G. Thorbergsson for very useful discussions, and specially the members of the organizing committee of the II Encuentro de Geometría Diferencial for their outstanding hospitality.
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Instituto de Matemática e Estatística,
Universidade de São Paulo
Rua do Matão, 1010
São Paulo, SP 05508-090 - Brazil
Recibido: 15 de octubre de 2005
Aceptado: 27 de septiembre de 2006