View source for Tight and taut immersions

The total absolute curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t0928101.png" /> of an immersion of differentiability class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t0928102.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t0928103.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t0928104.png" />, of a closed connected <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t0928105.png" />-manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t0928106.png" /> in Euclidean <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t0928107.png" />-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t0928108.png" /> (cf. [[Immersion of a manifold|Immersion of a manifold]]) is expressed as an integral (a2) in terms of local invariants. It obeys

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t0928109.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281010.png" /> is the infimum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281011.png" /> for all immersions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281012.png" /> into Euclidean space or for some special class of immersions, such as a component of the space of smooth imbeddings, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281013.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281014.png" />-th [[Betti number|Betti number]] for Čech homology with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281015.png" /> as coefficients (cf. [[Čech cohomology|Čech cohomology]]).

For a closed [[Curve|curve]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281017.png" /> is the curvature and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281018.png" /> the arc length. For a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281019.png" /> [[Surface|surface]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281020.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281023.png" /> is the [[Gaussian curvature|Gaussian curvature]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281024.png" /> the area form, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281025.png" /> the [[Euler characteristic|Euler characteristic]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281026.png" />. An immersion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281027.png" /> is called tight if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281028.png" /> and if this <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281029.png" /> has minimal total absolute curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281030.png" />, the lowest bound being attained. If, moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281031.png" /> lies in the unit sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281032.png" />, then the immersion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281033.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281034.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281035.png" /> is called taut. A taut immersion is always an imbedding.

The general definition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281036.png" /> is as follows:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281037.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>

Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281038.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281039.png" />-manifold and bundle of unit normal vectors for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281040.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281042.png" /> is the natural projection by the Gauss mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281043.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281044.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281045.png" /> is the volume element, an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281046.png" />-form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281047.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281048.png" /> the pull-back on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281049.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281050.png" /> the volume element on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281051.png" /> induced by the immersion into Euclidean space. The form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281052.png" /> is called the Lipschitz–Killing form. The well-defined density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281053.png" /> is the absolute Lipschitz–Killing density on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281054.png" />. For surfaces (in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281055.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281056.png" />. In general, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281057.png" /> measures the area swept out by the normal vectors on the unit sphere of directions. Many homogeneous spaces, like <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281058.png" />, all projective spaces and homogeneous Kähler manifolds, have tight (and even taut) imbeddings by their standard models in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281059.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281060.png" />. (See below.)

The main problem is existence. One is interested also in the special properties of tight immersions for a given manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281061.png" />.

Important is the following probabilistic definition:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281062.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>

Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281063.png" /> is a unit vector, the gradient of the linear function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281064.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281065.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281066.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281067.png" /> is a "height function on a manifoldheight function" on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281068.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281069.png" /> is the number of non-degenerate critical points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281070.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281071.png" /> the expectation (or mean) value for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281072.png" /> with respect to the standard invariant measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281073.png" />.

For smooth immersions one has

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281074.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a4)</td></tr></table>

The property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281075.png" />, another definition of tight, permits the application of [[Morse theory|Morse theory]]. The inequality (a1) is, in particular, a consequence of the Morse inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281076.png" />, which holds for almost-all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281077.png" />, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281078.png" /> is non-degenerate for almost-all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281079.png" />. It follows that an immersion is tight if every non-degenerate height function has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281080.png" /> critical points. See Fig. cand Fig. d.

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/t092810a.gif" />

Figure: t092810a

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/t092810b.gif" />

Figure: t092810b

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/t092810c.gif" />

Figure: t092810c

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/t092810d.gif" />

Figure: t092810d

An imbedding of spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281081.png" /> is called injective in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281083.png" />-homology if the induced homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281084.png" /> is injective for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281085.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281086.png" />, with as boundary the hyperplane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281087.png" />, be a half-space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281088.png" />. For example,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281089.png" /></td> </tr></table>

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281090.png" /> is a tight immersion and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281091.png" /> is a non-degenerate height function, then by Morse theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281092.png" /> is injective in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281093.png" />-homology. By continuity this injectivity then holds for every half-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281094.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281095.png" />. An example is the tight "Swiss cheese" , an imbedded surface with boundary, see Fig. e. A tight mapping into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281096.png" /> is also called a perfect function.

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/t092810e.gif" />

Figure: t092810e

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/t092810f.gif" />

Figure: t092810f

For curves and closed surfaces, the half-space property reduces to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281097.png" /> being connected for every half-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281098.png" />. Equivalent is Banchoff's two-piece property, which says that every hyperplane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t09281099.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810100.png" /> cuts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810101.png" /> 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|Projective geometry]]), as it is clearly invariant under any projective transformation in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810102.png" /> that sends the convex hull <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810103.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810104.png" />. Tautness as defined above is a conformal property (cf. [[Conformal geometry|Conformal geometry]]). It is invariant under any conformal (Möbius) transformation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810105.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810106.png" />, which, in turn, is determined by a unique projective transformation in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810107.png" /> sending <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810108.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810109.png" />.

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810110.png" />.

Miscellaneous representative theorems, mainly mentioned for surfaces, are as follows.

==Curves.==
A tight closed curve in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810111.png" /> is plane and convex (W. Fenchel, 1929). The plane curve in Fig. a is not tight by every definition. A knotted curve in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810112.png" /> like the trefoil knot in Fig. bhas <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810113.png" />. Equality with the infimum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810114.png" /> cannot be obtained (J. Milnor, 1950). Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810115.png" /> is the bridge index of the knot <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810116.png" />. It is the smallest number of maxima a height function can have on a knot <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810117.png" /> admitting isotopy of the knot (cf. [[Knot theory|Knot theory]]). The trefoil knot in Fig. ahas <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810118.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810119.png" />-manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810120.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810121.png" />, of differentiability class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810122.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810123.png" />, is a tight imbedding onto a convex hypersurface with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810124.png" />. The same conclusion is known for a continuous immersion with suitably defined <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810125.png" />.

==Surfaces.==
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810126.png" /> is a non-orientable closed surface with Euler number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810127.png" />, then no tight immersion into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810128.png" /> exists for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810129.png" /> ([[Projective plane|projective plane]]) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810130.png" /> (Klein bottle, cf. [[Klein surface|Klein surface]]), not even a continuous immersion. The case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810131.png" /> (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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810132.png" />. So, for every smooth immersion there exists a plane which cuts it in at least three pieces. All other surfaces have tight immersions into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810133.png" />. A tight torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810134.png" /> is depicted in Fig. cand a non-orientable tight surface with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810135.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810136.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810137.png" />, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810138.png" />, is necessarily an (algebraic) Veronese surface (topologically a real projective plane, cf. also [[Veronese mapping|Veronese mapping]]) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810139.png" />, unique up to projective transformations in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810140.png" /> (N.H. Kuiper, 1960).

T. Banchoff (1965) suggested, however, and W. Kühnel (1980, see [[#References|[a3]]]) proved, that except for the Klein bottle, a tight substantial polyhedral surface in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810141.png" /> exists exactly for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810142.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810143.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810144.png" />, is necessarily an imbedding into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810145.png" /> onto either the algebraic Veronese surface or onto Banchoff's six-vertex polyhedral surface [[#References|[a11]]]. Every smooth immersion of a surface with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810146.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810147.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810148.png" /> is regularly isotopic to a tight immersion (U. Pinkall, [[#References|[a15]]]). For the other surfaces the results are not yet complete. Every orientable surface with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810149.png" /> has a smooth substantial tight imbedding in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810150.png" />, but it

be analytic except for the torus (G. Thorbergsson, [[#References|[a19]]]). Every smooth imbedded knotted orientable surface in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810151.png" /> has total absolute curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810152.png" />, and equality cannot be attained if genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810153.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810154.png" />. For genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810155.png" />, however, Kuiper and W.F. Meeks [[#References|[a10]]] proved that there do exist "isotopy-tight" knotted surfaces with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810156.png" />. An example of a knotted surface of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810157.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810158.png" /> is depicted in Fig. e. It is obtained from two linked tight tori by two connecting handles of non-positive Gaussian curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810159.png" />. For this surface every non-degenerate height function has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810160.png" /> critical points.

Smooth immersions of surfaces in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810161.png" /> form a subclass of the smooth stable mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810162.png" />. In that class every surface has a tight stable mapping into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810163.png" />, hence with total absolute curvature equal to

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810164.png" /></td> </tr></table>

Tight analytic surfaces in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810165.png" /> are isometrically rigid in the class of analytic surfaces (A.D. Aleksandrov, 1938; see [[#References|[a3]]], p. 81, and [[Rigidity|Rigidity]]). Hardly anything more is known about <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810166.png" />-rigidity of non-convex smooth closed surfaces in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810167.png" />. However, by Kuiper's theorem (1955), no smooth closed surface in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810168.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810169.png" />-isometrically rigid. A surprising tight four-dimensional manifold in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810170.png" /> is Kühnel's topological imbedding of the complex projective plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810171.png" /> into the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810172.png" />-skeleton of a simplex in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810173.png" /> [[#References|[a9]]]. The image is a triangulation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810174.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810175.png" /> vertices.

Taut imbeddings deserve a separate discussion. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810176.png" /> be a compact connected space. The given extrinsic definition of tautness for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810177.png" /> in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810178.png" />-sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810179.png" />, namely by the property that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810180.png" /> is tight in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810181.png" />, evidently determines (with the half-space definition of tight) the following intrinsic definition. The subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810182.png" /> is taut in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810183.png" /> if the inclusion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810184.png" /> is injective in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810185.png" />-homology for every (round) ball <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810186.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810187.png" />. By stereographic projection from a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810188.png" /> into a Euclidean <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810189.png" />-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810190.png" /> orthogonal to the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810191.png" />, one obtains the following definition of tautness in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810192.png" />. A compact subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810193.png" /> is taut if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810194.png" /> contains no open set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810195.png" />, and the imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810196.png" /> is injective in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810197.png" />-homology, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810198.png" /> a round ball or the complement of a round ball in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810199.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810200.png" /> (cf. also [[Absolute retract for normal spaces|Absolute retract for normal spaces]]; [[Retract of a topological space|Retract of a topological space]]) is a smooth manifold. This is known for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810201.png" />. See [[#References|[a8]]].

The customary definitions of a smooth taut manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810202.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810203.png" /> are as follows: a) every non-degenerate distance function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810204.png" /> has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810205.png" /> critical points; and b) every round ball in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810206.png" /> meets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810207.png" /> in a subset that is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810208.png" />-homology injective in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810209.png" />. These definitions make sense and are also used for proper submanifolds of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810210.png" /> that are not necessarily compact (see [[#References|[a3]]]). But in that case tight is not defined and so tightness is not a consequence.

For a smooth proper submanifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810211.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810212.png" />, the customary requirement of tautness is a very strong condition. The only taut closed surfaces in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810213.png" /> are, up to a Möbius transformation, the following: homogeneous spaces, the round <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810214.png" />, the standard torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810215.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810216.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810217.png" /> are radii), and the standard Veronese surface (projective plane) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810218.png" />. Each of these models is a homogeneous space by motions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810219.png" />. Taut tori in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810220.png" /> are Dupin cyclides (cf. [[Dupin cyclide|Dupin cyclide]]). The diffeomorphism classes of all taut <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810221.png" />-manifolds in Euclidean spaces were found in [[#References|[a17]]].

T. Ozawa [[#References|[a14]]] proved that every connected set of critical points of a distance function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810222.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810223.png" /> on a closed taut manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810224.png" /> is itself a taut submanifold. The manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810225.png" /> 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|differential geometry]], in the study of the following kinds of spaces.

1) Orbits of isotropy representations of symmetric spaces, also called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810227.png" />-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 [[#References|[a1]]].

2) Closed isoparametric submanifolds. A compact submanifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810228.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810229.png" /> lies in a sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810230.png" />, but need not be homogeneous for codimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810231.png" />, [[#References|[a5]]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810232.png" /> is irreducible with codimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810233.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810234.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810235.png" />-space, [[#References|[a20]]]. Isoparametric submanifolds are taut. They form a generalization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810236.png" />-spaces and their cohomology can likewise be calculated from their associated marked Dynkin diagrams ([[#References|[a6]]]). The concepts of taut imbedding and isoparametric submanifold generalize to the Hilbert space setting [[#References|[a16]]]. Examples are the infinite-dimensional flag manifolds. Finally, a remarkable result due to H.-F. Münzer [[#References|[a12]]] is that for an isoparametric hypersurface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810237.png" /> in a sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810238.png" />, the number of distinct principal curvatures must be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810239.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810240.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810241.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810242.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810243.png" />.

3) A submanifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810244.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810245.png" /> is called totally focal if every distance function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810246.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810247.png" />) either has on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810248.png" /> 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 [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810249.png" />-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 [[#References|[a2]]]. The product of two taut imbeddings is taut, and cylinders and surfaces of revolution built from taut imbeddings are taut (see [[#References|[a3]]] and [[#References|[a15]]]). All closed taut submanifolds that are now known (1990) have been obtained by these and some other new constructions (see [[#References|[a18]]] and [[#References|[a13]]]). Perhaps these exhaust all possibilities. For a wealth of other results and generalizations see the references.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Bott, H. Samelson, "Applications of the theory of Morse to symmetric spaces" ''Amer. J. Math.'' , '''80''' (1958) pp. 964–1029 {{MR|0105694}} {{ZBL|0101.39702}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> T.E. Cecil, S.S. Chern, "Tautness and Lie sphere geometry" ''Math. Ann.'' , '''278''' (1987) pp. 381–399 {{MR|0909233}} {{ZBL|0635.53029}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> T.E. Cecil, P.J. Ryan, "Tight and taut immersions of manifolds" , Pitman (1985) {{MR|0781126}} {{ZBL|0596.53002}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> S. Carter, A. West, "Isoparametric and totally focal submanifolds" ''Proc. London Math. Soc.'' , '''60''' (1990) pp. 609–624 {{MR|1044313}} {{ZBL|0663.53045}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> D. Ferus, H. Karcher, H.F. Münzer, "Clifford Algebren und neue isoparametrische Hyperflächen" ''Math. Z.'' , '''177''' (1981) pp. 479–502 {{MR|624227}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> W.Y. Hsiang, R.S. Palais, C.L. Terng, "The topology of isoparametric submanifolds" ''J. Diff. Geom.'' , '''27''' (1988) pp. 423–460 {{MR|0940113}} {{ZBL|0618.57018}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> N.H. Kuiper, "Tight embeddings and maps" W.Y. Hsiang (ed.) et al. (ed.) , ''The Chern Symposium (1979)'' , Springer (1980) pp. 97–145 {{MR|0609559}} {{ZBL|0461.53033}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> N.H. Kuiper, "Taut sets in three-space are very special" ''Topology'' , '''23''' (1984) pp. 323–336 {{MR|0770568}} {{ZBL|0578.53044}} </TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> W. Kühnel, T. Banchoff, "The 9-vertex complex projective plane" ''The Math. Intelligencer'' , '''5''' : 3 (1983) pp. 11–22 {{MR|0737686}} {{ZBL|0534.51009}} </TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> N.H. Kuiper, W.F., III Meeks, "Total curvature for knotted surfaces" ''Invent. Math.'' , '''77''' (1984) pp. 25–69 {{MR|0751130}} {{ZBL|0553.53034}} </TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> N.H. Kuiper, W.F. Pohl, "Tight topological embedding of the real projective plane in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092810/t092810250.png" />" ''Invent. Math.'' , '''42''' (1977) pp. 177–199 {{MR|0494122}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top"> H.-F. Münzner, "Isoparametrische Hyperflächen in Sphären II" ''Math. Ann.'' , '''256''' (1981) pp. 215–232 {{MR|0620709}} {{ZBL|0438.53050}} </TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top"> R. Miyaoka, T. Ozawa, "Construction of taut embeddings and the Cecil–Ryan conjecture" , ''Proc. 1988 Symp. Differential geometry'' , Acad. Press (1990)</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top"> T. Ozawa, "On critical sets of distance functions to a taut submanifold" ''Math. Ann.'' , '''276''' (1986) pp. 91–96 {{MR|0863709}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top"> U. Pinkall, "Tight surfaces and regular homotopy" ''Topology'' , '''25''' (1986) pp. 475–481 {{MR|0862434}} {{ZBL|0605.53027}} </TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top"> R.S. Palais, C.L. Terng, "Critical point theory and submanifold geometry" , ''Lect. notes in math.'' , '''1353''' , Springer (1988) {{MR|0972503}} {{ZBL|0658.49001}} </TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top"> U. Pinkall, G. Thorbergsson, "Taut 3-manifolds" ''Topology'' , '''28''' (1989) pp. 389–402 {{MR|1030983}} {{ZBL|0686.53050}} </TD></TR><TR><TD valign="top">[a18]</TD> <TD valign="top"> U. Pinkall, G. Thorbergsson, "Deformations of Dupin hypersurfaces" ''Proc. Amer. Math. Soc.'' , '''107''' (1989) pp. 1037–1043 {{MR|0975655}} {{ZBL|0682.53061}} </TD></TR><TR><TD valign="top">[a19]</TD> <TD valign="top"> G. Thorbergsson, "Tight analytic surfaces" ''Topology'' (Forthcoming) {{MR|1113686}} {{ZBL|0727.57031}} </TD></TR><TR><TD valign="top">[a20]</TD> <TD valign="top"> G. Thorbergsson, "Isoparametric foliations and their buildings" ''Ann. of Math.'' , '''31''' (1991) pp. 429–446 {{MR|1097244}} {{ZBL|0727.57028}} </TD></TR></table>