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admits
Riemannian metrics of arbitrarily small volume, such that every essential surface is of area at least 1. Here a surface is called "essential" if it cannot be contracted to a point in the ambient 4-manifold.
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invariants. That is, systolic invariants or products of systolic invariants do not in general provide universal (i.e. curvature-free) lower bounds for the total volume of a closed
Riemannian manifold.
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The opposite of systolic freedom is systolic constraint, characterized by the presence of systolic inequalities such as
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Croke, Christopher B.; Katz, Mikhail (2003), "Universal volume bounds in
Riemannian manifolds",
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155:, Geom. Topol. Monogr., vol. 2, Coventry: Geom. Topol. Publ., pp. 113–123
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175:, Comput. Math. Ser., Boca Raton, FL: Chapman & Hall/CRC, pp. 287–320
67:). One of the first publications to study systolic freedom in detail is by
201:, Sémin. Congr., vol. 1, Paris: Soc. Math. France, pp. 291–362
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186:
Projective plane and planar quantum codesjournal=Found. Comput. Math.
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Actes de la Table Ronde de Géométrie Différentielle (Luminy, 1992)
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Berger, Marcel (1993), "Systoles et applications selon Gromov",
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Surveys in differential geometry, VIII (Boston, MA, 2002)
210:(1995), "Counterexamples to isosystolic inequalities",
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and others. Gromov's observation was elaborated on by
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Gromov's systolic inequality for essential manifolds
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Gromov's systolic inequality for essential manifolds
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197:(1996), "Systoles and intersystolic inequalities",
153:Proceedings of the Kirbyfest (Berkeley, CA, 1998)
356:Gromov's inequality for complex projective space
47:preprint in 1992 (which eventually appeared as
135:, Somerville, MA: Int. Press, pp. 109–137
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82:survey the main results on systolic freedom.
8:
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39:Systolic freedom was first detected by
171:-systolic freedom and quantum codes",
164:; Meyer, David A.; Luo, Feng (2002), "
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74:Systolic freedom has applications in
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127:216, Exp. No. 771, 5, 279–310.
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173:Mathematics of quantum computation
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51:), and was further developed by
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24:refers to the fact that closed
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28:may have arbitrarily small
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284:Loewner's torus inequality
184:; Meyer, David A. (2001),
92:complex projective plane
76:quantum error correction
318:1-systoles of manifolds
294:Filling area conjecture
80:Croke & Katz (2003)
277:1-systoles of surfaces
387:Differential geometry
18:differential geometry
304:Systoles of surfaces
182:Freedman, Michael H.
162:Freedman, Michael H.
151:-systolic-freedom",
142:Freedman, Michael H.
121:(in French), 1992/93
32:regardless of their
26:Riemannian manifolds
392:Riemannian geometry
309:Eisenstein integers
99:Systolic constraint
330:Essential manifold
224:10.1007/bf01264937
119:SĂ©minaire Bourbaki
397:Systolic geometry
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366:Systolic category
270:Systolic geometry
61:Marcel Berger
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361:Systolic freedom
340:Hermite constant
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53:Mikhail Katz
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69:Katz (1995)
49:Gromov 1996
381:Categories
125:Astérisque
111:References
144:(1999), "
232:11211702
45:I.H.É.S.
34:systolic
86:Example
63: (
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43:in an
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228:S2CID
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65:1993
220:doi
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