Tight and taut immersions
The total absolute curvature of an immersion of differentiability class , , , of a closed connected -manifold in Euclidean -space (cf. Immersion of a manifold) is expressed as an integral (a2) in terms of local invariants. It obeys
Here is the infimum of for all immersions of into Euclidean space or for some special class of immersions, such as a component of the space of smooth imbeddings, and is the -th Betti number for Čech homology with as coefficients (cf. Čech cohomology).
For a closed curve, , is the curvature and the arc length. For a surface in , , is the Gaussian curvature, the area form, and the Euler characteristic of . An immersion is called tight if and if this has minimal total absolute curvature , the lowest bound being attained. If, moreover, lies in the unit sphere , then the immersion of into is called taut. A taut immersion is always an imbedding.
The general definition of is as follows:
Here is the -manifold and bundle of unit normal vectors for at , is the natural projection by the Gauss mapping of into , is the volume element, an -form on , the pull-back on , and the volume element on induced by the immersion into Euclidean space. The form is called the Lipschitz–Killing form. The well-defined density is the absolute Lipschitz–Killing density on . For surfaces (in ), . In general, measures the area swept out by the normal vectors on the unit sphere of directions. Many homogeneous spaces, like , all projective spaces and homogeneous Kähler manifolds, have tight (and even taut) imbeddings by their standard models in for some . (See below.)
The main problem is existence. One is interested also in the special properties of tight immersions for a given manifold .
Important is the following probabilistic definition:
Here is a unit vector, the gradient of the linear function on with , is a "height function on a manifoldheight function" on , is the number of non-degenerate critical points of , and the expectation (or mean) value for with respect to the standard invariant measure on .
For smooth immersions one has
The property , another definition of tight, permits the application of Morse theory. The inequality (a1) is, in particular, a consequence of the Morse inequality , which holds for almost-all , as is non-degenerate for almost-all . It follows that an immersion is tight if every non-degenerate height function has critical points. See Fig. cand Fig. d.
An imbedding of spaces is called injective in -homology if the induced homomorphism is injective for . Let , with as boundary the hyperplane , be a half-space of . For example,
If is a tight immersion and is a non-degenerate height function, then by Morse theory is injective in -homology. By continuity this injectivity then holds for every half-space . For smooth immersions of closed manifolds this half-space property is equivalent to tightness. However, this half-space definition can be applied in the larger context of continuous immersions or even mappings of manifolds and other compact topological spaces into . An example is the tight "Swiss cheese" , an imbedded surface with boundary, see Fig. e. A tight mapping into is also called a perfect function.
For curves and closed surfaces, the half-space property reduces to being connected for every half-space . Equivalent is Banchoff's two-piece property, which says that every hyperplane in cuts in at most two connected pieces. See the tight surfaces in Fig. c, Fig. d, and a non-tight curve in Fig. b.
The half-space definition places tightness in classical geometry and convexity theory. Thus it follows that tightness is a projective property (cf. Projective geometry), as it is clearly invariant under any projective transformation in that sends the convex hull into . Tautness as defined above is a conformal property (cf. Conformal geometry). It is invariant under any conformal (Möbius) transformation of onto , which, in turn, is determined by a unique projective transformation in sending onto .
In proofs an important role is played by Kuiper's fundamental theorem. For imbeddings it says: Top sets of tightly imbedded spaces are tight. A top set is the intersection with a supporting half-space or hyperplane in .
Miscellaneous representative theorems, mainly mentioned for surfaces, are as follows.
A tight closed curve in is plane and convex (W. Fenchel, 1929). The plane curve in Fig. a is not tight by every definition. A knotted curve in like the trefoil knot in Fig. bhas . Equality with the infimum cannot be obtained (J. Milnor, 1950). Here is the bridge index of the knot . It is the smallest number of maxima a height function can have on a knot admitting isotopy of the knot (cf. Knot theory). The trefoil knot in Fig. ahas . On this knot every height function has at least two maxima but some height functions must have at least three by Milnor's bound above.
The first higher-dimensional theorem is due to S.S. Chern and J. Lashof (1957): A substantial (not in a hyper-plane) immersion of a closed -manifold , , of differentiability class with , is a tight imbedding onto a convex hypersurface with . The same conclusion is known for a continuous immersion with suitably defined .
If is a non-orientable closed surface with Euler number , then no tight immersion into exists for (projective plane) and (Klein bottle, cf. Klein surface), not even a continuous immersion. The case (projective plane with one handle) was an open problem since 1960. F. Haab proved (in 1990) that this surface has in fact no tight smooth immersion in Euclidean space . So, for every smooth immersion there exists a plane which cuts it in at least three pieces. All other surfaces have tight immersions into . A tight torus is depicted in Fig. cand a non-orientable tight surface with , in Fig. d. The following theorems show that higher codimension and analyticity drastically restrict the possibility and nature of tight immersions. Also, differentiability is restrictive in comparison with continuous or piecewise-linear immersions.
A smooth tight substantial closed surface in , for , is necessarily an (algebraic) Veronese surface (topologically a real projective plane, cf. also Veronese mapping) in , unique up to projective transformations in (N.H. Kuiper, 1960).
T. Banchoff (1965) suggested, however, and W. Kühnel (1980, see [a3]) proved, that except for the Klein bottle, a tight substantial polyhedral surface in exists exactly for . This number is Heawood's chromatic number, known from the map-colour theorem. The same upper bound seems to hold for continuous tight immersions.
In this context there is another remarkable theorem. A substantial tight continuous immersion of the real projective plane into , , is necessarily an imbedding into onto either the algebraic Veronese surface or onto Banchoff's six-vertex polyhedral surface [a11]. Every smooth immersion of a surface with or into is regularly isotopic to a tight immersion (U. Pinkall, [a15]). For the other surfaces the results are not yet complete. Every orientable surface with has a smooth substantial tight imbedding in , but it
be analytic except for the torus (G. Thorbergsson, [a19]). Every smooth imbedded knotted orientable surface in has total absolute curvature , and equality cannot be attained if genus or . For genus , however, Kuiper and W.F. Meeks [a10] proved that there do exist "isotopy-tight" knotted surfaces with . An example of a knotted surface of genus , is depicted in Fig. e. It is obtained from two linked tight tori by two connecting handles of non-positive Gaussian curvature . For this surface every non-degenerate height function has critical points.
Smooth immersions of surfaces in form a subclass of the smooth stable mappings . In that class every surface has a tight stable mapping into , hence with total absolute curvature equal to
Tight analytic surfaces in are isometrically rigid in the class of analytic surfaces (A.D. Aleksandrov, 1938; see [a3], p. 81, and Rigidity). Hardly anything more is known about -rigidity of non-convex smooth closed surfaces in . However, by Kuiper's theorem (1955), no smooth closed surface in is -isometrically rigid. A surprising tight four-dimensional manifold in is Kühnel's topological imbedding of the complex projective plane into the -skeleton of a simplex in [a9]. The image is a triangulation of with vertices.
Taut imbeddings deserve a separate discussion. Let be a compact connected space. The given extrinsic definition of tautness for in the -sphere , namely by the property that is tight in , evidently determines (with the half-space definition of tight) the following intrinsic definition. The subspace is taut in if the inclusion is injective in -homology for every (round) ball in . By stereographic projection from a point into a Euclidean -space orthogonal to the vector , one obtains the following definition of tautness in . A compact subspace is taut if and only if contains no open set of , and the imbedding is injective in -homology, with a round ball or the complement of a round ball in . Then taut implies tight. A taut subspace of the plane is either a circle or a round disc, from which an everywhere-dense union of disjoint open round discs is deleted (a "limit Swiss cheese" ). Banchoff's plane Swiss cheese in Fig. eis tight but not taut, although every circle does cut it in at most two pieces, but one piece could be not injective in homology. While excluding exotic examples by an ANR-assumption, it is conjectured that every compact taut absolute neighbourhood retract in (cf. also Absolute retract for normal spaces; Retract of a topological space) is a smooth manifold. This is known for . See [a8].
The customary definitions of a smooth taut manifold in are as follows: a) every non-degenerate distance function has critical points; and b) every round ball in meets in a subset that is -homology injective in . These definitions make sense and are also used for proper submanifolds of that are not necessarily compact (see [a3]). But in that case tight is not defined and so tightness is not a consequence.
For a smooth proper submanifold in , the customary requirement of tautness is a very strong condition. The only taut closed surfaces in are, up to a Möbius transformation, the following: homogeneous spaces, the round , the standard torus ( and are radii), and the standard Veronese surface (projective plane) in . Each of these models is a homogeneous space by motions of . Taut tori in are Dupin cyclides (cf. Dupin cyclide). The diffeomorphism classes of all taut -manifolds in Euclidean spaces were found in [a17].
T. Ozawa [a14] proved that every connected set of critical points of a distance function or on a closed taut manifold is itself a taut submanifold. The manifold then contains many low-dimensional taut submanifolds, like circles, and tends to be special for this reason. Tautness plays an important role in differential geometry, in the study of the following kinds of spaces.
1) Orbits of isotropy representations of symmetric spaces, also called -spaces (Kobayashi–Takeuchi), are taut submanifolds. They are, of course, homogeneous spaces and their cohomology was computed using (degenerate) tight height functions in a classic paper of R. Bott and H. Samelson [a1].
2) Closed isoparametric submanifolds. A compact submanifold is called isoparametric if it has a flat normal bundle and the principal curvatures in the direction of any parallel normal vector field are constant. Then lies in a sphere , but need not be homogeneous for codimension , [a5]. If is irreducible with codimension , then is an -space, [a20]. Isoparametric submanifolds are taut. They form a generalization of -spaces and their cohomology can likewise be calculated from their associated marked Dynkin diagrams ([a6]). The concepts of taut imbedding and isoparametric submanifold generalize to the Hilbert space setting [a16]. Examples are the infinite-dimensional flag manifolds. Finally, a remarkable result due to H.-F. Münzer [a12] is that for an isoparametric hypersurface in a sphere , the number of distinct principal curvatures must be , , , , or .
3) A submanifold in is called totally focal if every distance function () either has on all critical points non-degenerate or all critical points degenerate. In combined efforts over several years of T.E. Cecil and P.J. Ryan, S. Carter and A. West [a4], the latter finally obtained the result that closed totally focal manifolds are the same as closed isoparametric submanifolds.
Note that any Möbius transform, or stereographic projection, or tubular -neighbourhood boundary of a taut submanifold, like those mentioned above, is taut. Tautness is also invariant under the group of Lie sphere transformations, which contains the Möbius group as a subgroup [a2]. The product of two taut imbeddings is taut, and cylinders and surfaces of revolution built from taut imbeddings are taut (see [a3] and [a15]). All closed taut submanifolds that are now known (1990) have been obtained by these and some other new constructions (see [a18] and [a13]). Perhaps these exhaust all possibilities. For a wealth of other results and generalizations see the references.
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|[a13]||R. Miyaoka, T. Ozawa, "Construction of taut embeddings and the Cecil–Ryan conjecture" , Proc. 1988 Symp. Differential geometry , Acad. Press (1990)|
|[a14]||T. Ozawa, "On critical sets of distance functions to a taut submanifold" Math. Ann. , 276 (1986) pp. 91–96 MR0863709|
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|[a19]||G. Thorbergsson, "Tight analytic surfaces" Topology (Forthcoming) MR1113686 Zbl 0727.57031|
|[a20]||G. Thorbergsson, "Isoparametric foliations and their buildings" Ann. of Math. , 31 (1991) pp. 429–446 MR1097244 Zbl 0727.57028|
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