Tightness (differential geometry)

Tightness is a term from the mathematical branch of differential geometry , especially two-dimensional surface theory . A two-dimensional, compact , orientable surface without an edge is called tight if the integral of its absolute Gaussian curvature is as small as possible.

Introduction and definition

The ball with a “dent” has positive and negative curvature.

The convex hull of the sphere with a “dent” has a flat area where the curvature disappears.

Let it be a two-dimensional, compact, orientable surface without a border and denote the Gaussian curvature. After the Gauss-Bonnet is the surface integral of about equal to the times the Euler characteristic , that is ${\displaystyle M\subset \mathbb {R} ^{3}}$ ${\displaystyle K\colon M\rightarrow \mathbb {R} }$ ${\displaystyle K}$ ${\displaystyle M}$ ${\displaystyle 2\pi }$

${\displaystyle \int _{M}K\,\mathrm {d} A=2\pi \cdot \chi (M)}$.

Put and . Next, let the surface of the convex hull be from and the Gaussian curvature , which and is almost everywhere defined . Then ${\displaystyle M_{+}:=\{x\in M\mid K(x)>0\}}$ ${\displaystyle M_{-}:=M\setminus M_{+}}$ ${\displaystyle {\tilde {M}}}$ ${\displaystyle M}$ ${\displaystyle {\tilde {K}}}$ ${\displaystyle {\tilde {M}}}$ ${\displaystyle \geq 0}$

${\displaystyle \int _{\tilde {M}}{\tilde {K}}\,\mathrm {d} A=2\pi \cdot 2=4\pi }$

and must disappear (here some details are ignored, since the convex hull has only weak differentiability properties). thats why ${\displaystyle {\tilde {K}}}$ ${\displaystyle {\tilde {M}}\setminus M_{+}}$

${\displaystyle \int _{M_{+}}K\,\mathrm {d} A\geq \int _{M_{+}\cap {\tilde {M}}}K\,\mathrm {d} A=\int _{\tilde {M}}{\tilde {K}}\,\mathrm {d} A=4\pi }$

and the simple calculation

${\displaystyle {\begin{aligned}\int _{M}|K|\,\mathrm {d} A&=\int _{M_{+}}K\,\mathrm {d} A-\int _{M_{-}}K\,\mathrm {d} A\\&=2\cdot \int _{M_{+}}K\,\mathrm {d} A-\left(\int _{M_{+}}K\,\mathrm {d} A+\int _{M_{-}}K\,\mathrm {d} A\right)\\&=2\cdot \int _{M_{+}}K\,\mathrm {d} A-\int _{M}K\,\mathrm {d} A\\&\geq 2\cdot 4\pi -2\pi \cdot \chi (M)\\&=2\pi \cdot (4-\chi (M))\end{aligned}}}$

gives a lower bound for the integral of the absolute curvature over . ${\displaystyle |K|}$ ${\displaystyle M}$

A two-dimensional, compact, orientable surface without an edge is called tight if this lower bound is accepted, that is, if ${\displaystyle M\subset \mathbb {R} ^{3}}$

${\displaystyle \int _{M}|K|\,\mathrm {d} A=2\pi \cdot (4-\chi (M))}$.

Equivalent characterizations

For a two-dimensional, compact, orientable surface without a border, the following statements are equivalent: ${\displaystyle M\subset \mathbb {R} ^{3}}$

The red plane divides the surface of the dented sphere into three parts, the bulge above and the two bowl-shaped parts below. So this area is not tight.

• ${\displaystyle M}$is tight, that is . ${\displaystyle \int _{M}|K|\,\mathrm {d} A=2\pi \cdot (4-\chi (M))}$
• ${\displaystyle \int _{M_{+}}K\,\mathrm {d} A=4\pi }$
• Each level is broken down into at most two connected components , that is, if and the two open half-spaces with , then are empty or connected for each . ${\displaystyle E\subset \mathbb {R} ^{3}}$ ${\displaystyle M}$ ${\displaystyle H_{1}}$ ${\displaystyle H_{2}}$ ${\displaystyle M\setminus E=H_{1}\cup H_{2}}$ ${\displaystyle M\cap H_{i}}$ ${\displaystyle i\in \{1,2\}}$

The third property is called the two-piece property or TPP for short, after the English term two-piece property . This equivalent property does not use differential geometric terms and therefore allows the tightness to be generalized to more general surfaces. Obviously, the surface of every convex set has the TPP. One can therefore view tightness as a generalization of convexity.

Examples

The spherical surface with radius is known to have constant Gaussian curvature and Euler characteristic 2. Therefore, ${\displaystyle r}$ ${\displaystyle {\tfrac {1}{r^{2}}}}$

${\displaystyle \int _{M}|K|\,\mathrm {d} A=\int _{M}{\frac {1}{r^{2}}}\,\mathrm {d} A={\frac {1}{r^{2}}}\cdot 4\pi r^{2}=4\pi =2\pi \cdot 2=2\pi \cdot (4-\chi (M))}$.

The spherical surface is therefore tight. This is much easier to see using the TPP since the sphere is convex, because apparently every convex surface has the TPP.

The torus surface with radii and that go through ${\displaystyle R}$ ${\displaystyle r}$

${\displaystyle F\colon [0,2\pi ]^{2}\rightarrow \mathbb {R} ^{3},\quad (\theta ,\varphi )\mapsto {\begin{pmatrix}(R+r\cos(\theta ))\cos(\varphi )\\(R+r\cos(\theta ))\sin(\varphi )\\r\sin(\varphi ))\end{pmatrix}}}$

is parameterized, has the curvature at that point ${\displaystyle F(\theta ,\varphi )}$

The curvature of the torus is positive towards the outside and negative towards the inside.

${\displaystyle K(F(\theta ,\varphi ))={\frac {\cos(\theta )}{r(R+r\cos(\theta ))}}}$

and the Euler characteristic of the torus surface is 0. Then one calculates

${\displaystyle \int _{M}|K|\,\mathrm {d} A=\int _{0}^{2\pi }\int _{0}^{2\pi }{\frac {|{\cos(\theta )}|}{r(R+r\cos(\theta ))}}\cdot {\sqrt {g(\theta ,\varphi )}}\,\mathrm {d} \theta \mathrm {d} \varphi }$

With   ${\displaystyle g(\theta ,\varphi )=\mathrm {det} ((DF)^{T}\cdot DF)(\theta ,\varphi )=r^{2}(R+r\cos(\theta ))^{2}}$

${\displaystyle {\begin{aligned}\qquad \qquad \;&=\int _{0}^{2\pi }\int _{0}^{2\pi }{\frac {|{\cos(\theta )}|}{r(R+r\cos(\theta ))}}\cdot r(R+r\cos(\theta ))\,\mathrm {d} \theta \mathrm {d} \varphi =\int _{0}^{2\pi }\int _{0}^{2\pi }|{\cos(\theta )}|\,\mathrm {d} \theta \mathrm {d} \varphi \\&=2\pi \cdot \int _{0}^{2\pi }|{\cos(\theta )}|\,\mathrm {d} \theta =2\pi \cdot 4=2\pi \cdot (4-\chi (M)).\end{aligned}}}$

Hence, the torus surface is an example of a non-convex tight surface.

For each gender there are two-dimensional, compact, orientable, borderless surfaces that are tight. ${\displaystyle M\subset \mathbb {R} ^{3}}$

Tight immersions

Let it be an immersion of a differentiable, two-dimensional, orientable manifold in the three-dimensional Euclidean space. For is a unit vector that is perpendicular to the tangential plane in , so that is continuous (for this one needs the orientability). The Gaussian curvature is defined as the determinant of the Jacobian matrix of the figure . Then you can make very similar considerations as above and say tight if ${\displaystyle f\colon M\rightarrow \mathbb {R} ^{3}}$ ${\displaystyle M}$ ${\displaystyle x\in M}$ ${\displaystyle \nu _{f}(x)\in \mathbb {R} ^{3}}$ ${\displaystyle f(x)}$ ${\displaystyle x\mapsto \nu _{f}(x)}$ ${\displaystyle K_{f}(x)}$ ${\displaystyle \nu _{f}\colon M\rightarrow S^{2}}$ ${\displaystyle f}$

${\displaystyle \int _{M}|K_{f}|\,\mathrm {d} A=2\pi \cdot (4-\chi (M))}$.

The tightness of a surface defined above means the tightness of immersion . Of course, there are other immersions that are tight, about the restriction of the linear mapping with real constants on which evidently an immersion of the spherical surface to an ellipsoid in is. This immersion is also tight. On the other hand, the immersion of the indented spherical surface is not tight, nor is the immersion that depicts the indentation. ${\displaystyle M\subset \mathbb {R} ^{3}}$ ${\displaystyle \mathrm {id} _{M}\colon M\rightarrow M\subset R^{3}}$ ${\displaystyle \mathbb {R} ^{3}\rightarrow \mathbb {R} ^{3},\,(x,y,z)\mapsto (ax,by,cz)}$ ${\displaystyle a,b,c>0}$ ${\displaystyle S^{2}}$ ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle \mathrm {id} _{M}}$ ${\displaystyle M}$ ${\displaystyle S^{2}\rightarrow M\subset \mathbb {R} ^{3}}$

In this connection the following theorem of Chern and Lashof applies : Every tight immersion of the spherical surface in the maps onto the surface of a convex set . ${\displaystyle \mathbb {R} ^{3}}$

In the cited textbook "Tight and taut immersions of manifolds" by TE Cecil and PJ Ryan there is a systematic investigation of tight immersions and related terms. There further equivalent characterizations using properties of the mapping as well as generalizations to higher dimensions are dealt with, with the TPP (two-piece property) also playing an important role. ${\displaystyle f\colon M\rightarrow \mathbb {R} ^{3}}$

Individual evidence

1. ↑ Wolfgang Kühnel : Differentialgeometrie , ISBN 978-3-8348-0411-2 , definition 4.47
2. ↑ Wolfgang Kühnel: Differentialgeometrie , ISBN 978-3-8348-0411-2 , consequence 4.48
3. ↑ Thomas Banchoff , Wolfgang Kühnel: Tight Submanifolds, Smooth and Polyhedral , MSRI publications, Volume 32, Cambridge University Press (1997), ISBN 0-521-62047-3 , pages 52-118
4. ↑ Tevian Dray: Differential Forms and the Geometry of General Relativity , CRC Press (2015), chap. 18: Curvature , formula (18.88) on page 232
5. ↑ Manfredo P. do Carmo : Differential geometry of curves and surfaces , Vieweg-Verlag, 3rd edition (1993), ISBN 978-3-528-27255-5 , paragraph 3.3: The Gaussian mapping in local coordinates , page 116
6. ↑ Thomas Banchoff, Nicolaas Kuiper: Geometrical class and degree for surfaces in three-space , Journal of Differential Geometry (1981), Volume 16, pages 559-576
7. ^ TE Cecil, PJ Ryan: Tight and taut immersions of manifolds , Pitman Publishing Inc. (1985), ISBN 0-273-08631-6 , definition on page 2
8. ^ TE Cecil, PJ Ryan: Tight and taut immersions of manifolds , Pitman Publishing Inc. (1985), ISBN 0-273-08631-6 , Theorem 7.16