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.
Figure: t092810a
Figure: t092810b
Figure: t092810c
Figure: t092810d
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.
Figure: t092810e
Figure: t092810f
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.
Curves.
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
.
Surfaces.
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.