Introduction to systolic geometry

<h2 id="surface-tension-and-shape-of-a-water-drop">Surface tension and shape of a water drop</h2>

<p>Perhaps the most familiar physical manifestation of the 3-dimensional isoperimetric inequality is the shape of a drop of water.  Namely, a drop will typically assume a symmetric round shape.  Since the amount of water in a drop is fixed, surface tension forces the drop into a shape which minimizes the surface area of the drop, namely a round sphere.  Thus the round shape of the drop is a consequence of the phenomenon of surface tension.  Mathematically, this phenomenon is expressed by the isoperimetric inequality.
</p>
<h2 id="isoperimetric-inequality-in-the-plane">Isoperimetric inequality in the plane</h2>
<p>The solution to the isoperimetric problem in the plane is usually expressed in the form of an inequality that relates the length 
  
    
      
        L
      
    
    {\displaystyle L}
  
 of a closed curve and the area 
  
    
      
        A
      
    
    {\displaystyle A}
  
 of the planar region that it encloses. The isoperimetric inequality states that
</p>

4
        π
        A
        ≤
        
          L
          
            2
          
        
        ,
        
      
    
    {\displaystyle 4\pi A\leq L^{2},\,}

<p>and that the equality holds if and only if the curve is a round circle.  The inequality is an upper bound for area in terms of length.
</p>
<h2 id="central-symmetry">Central symmetry</h2>
<p>Recall the notion of central symmetry: a Euclidean polyhedron is called centrally symmetric if it is invariant under the <a href="/facts/Antipodal_map/2V3228jA">antipodal map</a>
</p>

x
        ↦
        −
        x
        .
        
      
    
    {\displaystyle x\mapsto -x.\,}

<p>Thus, in the plane central symmetry is the rotation by 180 degrees.  For example, an ellipse is centrally symmetric, as is any ellipsoid in 3-space.
</p>
<h2 id="property-of-a-centrally-symmetric-polyhedron-in-3-space">Property of a centrally symmetric polyhedron in 3-space</h2>
<p>There is a geometric inequality that is in a sense dual to the isoperimetric inequality in the following sense.  Both involve a length and an area.  The isoperimetric inequality is an upper bound for area in terms of length.  There is a geometric inequality which provides an upper bound for a certain length in terms of area.  More precisely it can be described as follows.
</p><p>Any centrally symmetric convex body of surface area 
  
    
      
        A
      
    
    {\displaystyle A}
  
 can be squeezed through a noose of length 
  
    
      
        
          
            π
            A
          
        
      
    
    {\displaystyle {\sqrt {\pi A}}}
  
, with the tightest fit achieved by a sphere.  This property is equivalent to a special case of Pu's inequality, one of the earliest systolic inequalities.
</p><p>For example, an ellipsoid is an example of a convex centrally symmetric body in 3-space.  It may be helpful to the reader to develop an intuition for the property mentioned above in the context of thinking about ellipsoidal examples.
</p><p>An alternative formulation is as follows.  Every convex centrally symmetric body 
  
    
      
        P
      
    
    {\displaystyle P}
  
 in 
  
    
      
        
          
            
              R
            
          
          
            3
          
        
      
    
    {\displaystyle {\mathbb {R} }^{3}}
  
 admits a pair of opposite (antipodal) points and a path of length

L
      
    
    {\displaystyle L}
  
 joining them and lying on the boundary 
  
    
      
        ∂
        P
      
    
    {\displaystyle \partial P}
  
 of 
  
    
      
        P
      
    
    {\displaystyle P}
  
, satisfying
</p>

L
          
            2
          
        
        ≤
        
          
            π
            4
          
        
        
          a
          r
          e
          a
        
        (
        ∂
        P
        )
        .
      
    
    {\displaystyle L^{2}\leq {\frac {\pi }{4}}\mathrm {area} (\partial P).}

<h2 id="notion-of-systole">Notion of systole</h2>

<p>The <i>systole</i> of a compact metric space 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is a metric
invariant of 
  
    
      
        X
      
    
    {\displaystyle X}
  
, defined to be the least length of a
noncontractible loop in 
  
    
      
        X
      
    
    {\displaystyle X}
  
.  We will denote it as follows:
</p>

s
          y
          s
        
        (
        X
        )
        .
        
      
    
    {\displaystyle \mathrm {sys} (X).\,}

<p>Note that a loop minimizing length is necessarily a <a href="/facts/Closed_geodesic/DlBzcCgT">closed geodesic</a>.  When 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is a <a href="/facts/Graph_(discrete_mathematics)/kw3eIBUe">graph</a>, the invariant is usually referred to as the <a href="/facts/Girth_(graph_theory)/rPwRGgxj">girth</a>, ever since the 1947 article by <a href="/facts/William_Tutte/8VbfbzU6">William Tutte</a>. Possibly inspired by Tutte's article, <a href="/facts/Charles_Loewner/caA1Feuu">Charles Loewner</a> started thinking about systolic questions on surfaces in the late 1940s, resulting in a 1950 thesis by his student P. M. Pu.  The actual term <i>systole</i> itself was not coined until a quarter century later, by <a href="/facts/Marcel_Berger/bgaDdaU1">Marcel Berger</a>.
</p><p>This line of research was, apparently, given further impetus by a remark of <a href="/facts/Ren%C3%A9_Thom/hdzhWSaC">René Thom</a>, in a conversation with Berger in the library of Strasbourg University during the 1961–62 academic year, shortly after the publication of the papers of R. Accola and C. Blatter. Referring to these systolic inequalities, Thom reportedly exclaimed: <i>Mais c'est fondamental!</i>  [These results are of fundamental importance!]
</p><p>Subsequently, Berger popularized the subject in a series of articles and books, most recently in the March 2008 issue of the <a href="/facts/Notices_of_the_American_Mathematical_Society/smHYV3sv">Notices of the American Mathematical Society</a>.  A bibliography at the <i>Website for systolic geometry and topology</i> currently contains over 170 articles.  Systolic geometry is a rapidly developing field, featuring a number of recent publications in leading journals. Recently, an intriguing link has emerged with the <a href="/facts/Lusternik%E2%80%93Schnirelmann_category/grHpYbmN">Lusternik–Schnirelmann category</a>.  The existence of such a link can be thought of as a theorem in systolic topology.
</p>
<h2 id="the-real-projective-plane">The real projective plane</h2>

<p>In <a href="/facts/Projective_geometry/rHKSaEbx">projective geometry</a>, the <a href="/facts/Real_projective_plane/gGVd0ctY">real projective plane</a> 
  
    
      
        
          
            R
            P
          
          
            2
          
        
      
    
    {\displaystyle \mathbb {RP} ^{2}}
  
 is defined as the collection of lines through the origin in 
  
    
      
        
          
            R
          
          
            3
          
        
      
    
    {\displaystyle \mathbb {R} ^{3}}
  
.  The distance function on 
  
    
      
        
          
            R
            P
          
          
            2
          
        
      
    
    {\displaystyle \mathbb {RP} ^{2}}
  
 is most readily understood from this point of view.  Namely, the distance between two lines through the origin is by definition the angle between them (measured in radians), or more precisely the lesser of the two angles.  This distance function corresponds to the metric of constant <a href="/facts/Gaussian_curvature/ARdhnb7o">Gaussian curvature</a> +1.
</p><p>Alternatively, 
  
    
      
        
          
            R
            P
          
          
            2
          
        
      
    
    {\displaystyle \mathbb {RP} ^{2}}
  
 can be defined as the surface obtained by identifying each pair of antipodal points on the 2-sphere.
</p><p>Other metrics on 
  
    
      
        
          
            R
            P
          
          
            2
          
        
      
    
    {\displaystyle \mathbb {RP} ^{2}}
  
 can be obtained by quotienting metrics on 
  
    
      
        
          S
          
            2
          
        
      
    
    {\displaystyle S^{2}}
  
 imbedded in 3-space in a centrally symmetric way.
</p><p>Topologically, 
  
    
      
        
          
            R
            P
          
          
            2
          
        
      
    
    {\displaystyle \mathbb {RP} ^{2}}
  
 can be obtained from the Möbius strip by attaching a disk along the boundary.
</p><p>Among <a href="/facts/Closed_surface/jK1b1KIG">closed surfaces</a>, the real projective plane is the simplest non-orientable such surface.
</p>
<h2 id="pus-inequality">Pu's inequality</h2>
<p><a href="/facts/Pu%27s_inequality/vDc8bw6E">Pu's inequality for the real projective plane</a> applies to general <a href="/facts/Riemannian_metric/5KYnmEgF">Riemannian metrics</a> on 
  
    
      
        
          
            R
            P
          
          
            2
          
        
      
    
    {\displaystyle \mathbb {RP} ^{2}}
  
.
</p><p>A student of <a href="/facts/Charles_Loewner/caA1Feuu">Charles Loewner</a>'s, <a href="/facts/Pao_Ming_Pu/CNIJsHBH">Pao Ming Pu</a> proved in a 1950 thesis (published in 1952) that every metric 
  
    
      
        g
      
    
    {\displaystyle g}
  
 on the real projective plane 
  
    
      
        
          
            R
            P
          
          
            2
          
        
      
    
    {\displaystyle \mathbb {RP} ^{2}}
  
 satisfies the optimal inequality
</p>

s
          y
          s
        
        (
        g
        
          )
          
            2
          
        
        ≤
        
          
            π
            2
          
        
        
          a
          r
          e
          a
        
        (
        g
        )
        ,
      
    
    {\displaystyle \mathrm {sys} (g)^{2}\leq {\frac {\pi }{2}}\mathrm {area} (g),}

<p>where 
  
    
      
        
          s
          y
          s
        
      
    
    {\displaystyle \mathrm {sys} }
  
 is the systole.  The boundary case of equality is attained precisely when the metric is of constant Gaussian curvature.  Alternatively, the inequality can be presented as follows:
</p>

a
          r
          e
          a
        
        (
        g
        )
        −
        
          
            2
            π
          
        
        
          s
          y
          s
        
        (
        g
        
          )
          
            2
          
        
        ≥
        0.
      
    
    {\displaystyle \mathrm {area} (g)-{\frac {2}{\pi }}\mathrm {sys} (g)^{2}\geq 0.}

<p>There is a vast generalisation of Pu's inequality, due to <a href="/facts/Mikhail_Gromov_(mathematician)/N0t5hhcI">Mikhail Gromov</a>, called <a href="/facts/Gromov%27s_systolic_inequality_for_essential_manifolds/nVMmVnPA">Gromov's systolic inequality for essential manifolds</a>.  To state his result, one requires a topological notion of an <a href="/facts/Essential_manifold/PKBNxMGR">essential manifold</a>.
</p>
<h2 id="loewners-torus-inequality">Loewner's torus inequality</h2>

<p>Similarly to Pu's inequality, <a href="/facts/Loewner%27s_torus_inequality/pkF4mAxv">Loewner's torus inequality</a> relates
the total area, to the systole, i.e. least length of a noncontractible
loop on the torus 
  
    
      
        (
        
          T
          
            2
          
        
        ,
        g
        )
      
    
    {\displaystyle (T^{2},g)}
  
:
</p>

a
          r
          e
          a
        
        (
        g
        )
        −
        
          
            
              
                3
              
              2
            
          
        
        
          s
          y
          s
        
        (
        g
        
          )
          
            2
          
        
        ≥
        0.
      
    
    {\displaystyle \mathrm {area} (g)-{\tfrac {\sqrt {3}}{2}}\mathrm {sys} (g)^{2}\geq 0.}

<p>The boundary case of equality is attained if and only if the metric is
homothetic to the flat metric obtained as the quotient of

R
            
          
          
            2
          
        
      
    
    {\displaystyle {\mathbb {R} }^{2}}
  
 by the lattice formed by the
<a href="/facts/Eisenstein_integers/UGQmFI8c">Eisenstein integers</a>.
</p>
<h2 id="bonnesens-inequality">Bonnesen's inequality</h2>
<p>The classical <a href="/facts/Bonnesen%27s_inequality/NHWeMTqy">Bonnesen's inequality</a> is the strengthened
isoperimetric inequality
</p>

L
          
            2
          
        
        −
        4
        π
        A
        ≥
        
          π
          
            2
          
        
        (
        R
        −
        r
        
          )
          
            2
          
        
        .
        
      
    
    {\displaystyle L^{2}-4\pi A\geq \pi ^{2}(R-r)^{2}.\,}

<p>Here 
  
    
      
        A
      
    
    {\displaystyle A}
  
 is the area of the region bounded by a closed Jordan curve of length (perimeter) 
  
    
      
        L
      
    
    {\displaystyle L}
  
 in the plane, 
  
    
      
        R
      
    
    {\displaystyle R}
  
 is the circumradius of the bounded region, and 
  
    
      
        r
      
    
    {\displaystyle r}
  
 is its inradius.  The error term 
  
    
      
        
          π
          
            2
          
        
        (
        R
        −
        r
        
          )
          
            2
          
        
      
    
    {\displaystyle \pi ^{2}(R-r)^{2}}
  
 on the right hand side is traditionally called the <i>isoperimetric defect</i>.  There exists a similar strengthening of Loewner's inequality.
</p>
<h2 id="loewners-inequality-with-a-defect-term">Loewner's inequality with a defect term</h2>
<p>The explanation of the strengthened version of Loewner's inequality is somewhat more technical than the rest of this article.  It seems worth including it here for the sake of completeness.  The strengthened version is the inequality
</p>

a
          r
          e
          a
        
        (
        g
        )
        −
        
          
            
              
                3
              
              2
            
          
        
        
          s
          y
          s
        
        (
        g
        
          )
          
            2
          
        
        ≥
        
          V
          a
          r
        
        (
        f
        )
        ,
      
    
    {\displaystyle \mathrm {area} (g)-{\tfrac {\sqrt {3}}{2}}\mathrm {sys} (g)^{2}\geq \mathrm {Var} (f),}

<p>where Var is the probabilistic <a href="/facts/Variance/ULBJKXD1">variance</a> while <i>f</i> is the conformal factor expressing the metric <i>g</i> in terms of the flat metric of unit area in the conformal class of <i>g</i>.  The proof results from a combination of the computational formula for the variance and <a href="/facts/Fubini%27s_theorem/q4lhtS3s">Fubini's theorem</a> (see Horowitz <i>et al</i>, 2009).
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Systoles_of_surfaces/dIVJqvsL">Systoles of surfaces</a></li>
<li><a href="/facts/Sub-Riemannian_geometry/TVnHsBSo">Sub-Riemannian geometry</a></li></ul>

<ul><li><a href="/facts/Victor_Bangert/au51wtp4">Bangert, V.</a>; Croke, C.; Ivanov, S.; Katz, M. (2005). "Filling area conjecture and ovalless real hyperelliptic surfaces". <i>Geometric and Functional Analysis</i>. 15 (3): 577–597. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/math/0405583">math/0405583</a>. <a href="/facts/CiteSeerX_(identifier)/SceDmd3c">CiteSeerX</a> <a href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.240.2242">10.1.1.240.2242</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2Fs00039-005-0517-8">10.1007/s00039-005-0517-8</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:17100812">17100812</a>.</li>
<li>Berger, Marcel (1992–1993). <a href="http://www.numdam.org/article/SB_1992-1993__35__279_0.pdf">"Systoles et applications selon Gromov"</a> (PDF). <i>Séminaire Bourbaki</i>. 35: 279–310.</li>
<li>Berger, M. (2003). <a href="https://books.google.com/books?id=yOclBQAAQBAJ&pg=PR11"><i>A panoramic view of Riemannian geometry</i></a>. Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-642-18245-7.</li>
<li>Berger, M. (2008). <a href="https://www.ams.org/notices/200803/tx080300374p.pdf">"What is... a Systole?"</a> (PDF). <i>Notices of the AMS</i>. 55 (3): 374–6.</li>
<li>Buser, P.; Sarnak, P. (1994). "On the period matrix of a Riemann surface of large genus. With an appendix by J.H. Conway and N.J.A. Sloane". <i>Invent. Math</i>. 117 (1): 27–56. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1994InMat.117...27B">1994InMat.117...27B</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF01232233">10.1007/BF01232233</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:116904696">116904696</a>.</li>
<li>Gromov, M. (1983). "Filling Riemannian manifolds". <i>J. Diff. Geom</i>. 18: 1–147. <a href="/facts/CiteSeerX_(identifier)/SceDmd3c">CiteSeerX</a> <a href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.400.9154">10.1.1.400.9154</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.4310%2Fjdg%2F1214509283">10.4310/jdg/1214509283</a>.</li>
<li>Gromov, M. (1996). "Systoles and intersystolic inequalities". <i>Actes de la Table Ronde de Géométrie Différentielle (Luminy, 1992)</i>. Sémin. Congr. Vol. 1. Soc. Math. France. pp. 291–362. <a href="/facts/CiteSeerX_(identifier)/SceDmd3c">CiteSeerX</a> <a href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.539.1365">10.1.1.539.1365</a>.</li>
<li>Horowitz, Charles; Katz, Karin Usadi; Katz, Mikhail G. (2009). "Loewner's torus inequality with isosystolic defect". <i>Journal of Geometric Analysis</i>. 19 (4): 796–808. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/0803.0690">0803.0690</a>. <a href="/facts/CiteSeerX_(identifier)/SceDmd3c">CiteSeerX</a> <a href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.314.5106">10.1.1.314.5106</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2Fs12220-009-9090-y">10.1007/s12220-009-9090-y</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:18444111">18444111</a>.</li>
<li>Katz, M.; Semmes, S.; Gromov, M. (2007) [2001]. <a href="/facts/Metric_Structures_for_Riemannian_and_Non-Riemannian_Spaces/DWTDq672"><i>Metric Structures for Riemannian and Non-Riemannian Spaces</i></a>. Progress in Mathematics. Vol. 152. Birkhäuser. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-8176-4583-0.</li>
<li><a href="/facts/Mikhail_Katz/2Fy17Yyy">Katz, M.</a> (1983). <a href="https://doi.org/10.4310%2Fjdg%2F1214437785">"The filling radius of two-point homogeneous spaces"</a>. <i><a href="/facts/Journal_of_Differential_Geometry/X7rU5sdg">Journal of Differential Geometry</a></i>. 18 (3): 505–511. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.4310%2Fjdg%2F1214437785">10.4310/jdg/1214437785</a>.</li>
<li>Katz, M. (2007). <a href="https://books.google.com/books?id=R5_zBwAAQBAJ&pg=PR7"><i>Systolic geometry and topology</i></a>. Mathematical Surveys and Monographs. Vol. 137. <a href="/facts/American_Mathematical_Society/fr5gotCt">American Mathematical Society</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-8218-4177-8.</li>
<li>Katz, M.; Rudyak, Y. (2006). "Systolic category and Lusternik–Schnirelman category of low-dimensional manifolds". <i>Communications on Pure and Applied Mathematics</i>. 59: 1433–56. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/math/0410456">math/0410456</a>. <a href="/facts/CiteSeerX_(identifier)/SceDmd3c">CiteSeerX</a> <a href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.236.3757">10.1.1.236.3757</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1002%2Fcpa.20146">10.1002/cpa.20146</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:15470409">15470409</a>.</li>
<li>Katz, M.; Sabourau, S. (2005). "Entropy of systolically extremal surfaces and asymptotic bounds". <i>Ergo. Th. Dynam. Sys</i>. 25 (4): 1209–20. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/math/0410312">math/0410312</a>. <a href="/facts/CiteSeerX_(identifier)/SceDmd3c">CiteSeerX</a> <a href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.236.5949">10.1.1.236.5949</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1017%2FS0143385704001014">10.1017/S0143385704001014</a> (inactive 2024-11-21). <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:11631690">11631690</a>.{{cite journal}}:  CS1 maint: DOI inactive as of November 2024 (link)</li>
<li>Katz, M.; Schaps, M.; Vishne, U. (2007). "Logarithmic growth of systole of arithmetic Riemann surfaces along congruence subgroups". <i>J. Differential Geom</i>. 76 (3): 399–422. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/math.DG/0505007">math.DG/0505007</a>. <a href="/facts/CiteSeerX_(identifier)/SceDmd3c">CiteSeerX</a> <a href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.240.5600">10.1.1.240.5600</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.4310%2Fjdg%2F1180135693">10.4310/jdg/1180135693</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:18152345">18152345</a>.</li>
<li>Pu, P.M. (1952). <a href="https://msp.org/pjm/1952/2-1/pjm-v2-n1-s.pdf#page=57">"Some inequalities in certain nonorientable Riemannian manifolds"</a> (PDF). <i>Pacific J. Math</i>. 2: 55–71. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2140%2Fpjm.1952.2.55">10.2140/pjm.1952.2.55</a>.</li></ul>

<h2 id="external-links">External links</h2>
<ul><li>Waner, Stefan (May 2014). <a href="https://www.zweigmedia.com/pdfs/DiffGeom.pdf">"Introduction to Differential Geometry & General Relativity"</a> (PDF). <i>Lecture Notes</i>. Departments of Mathematics and Physics, Hofstra University.</li></ul>

Introduction to systolic geometry open-in-new

Introduction to systolic geometry