What’s systole? Mijia Lai 2019.01.10
Definition The Systole of a compact metric space X is the least length of a non-contractible loop in X, denoted by sys π 1 . 1
Definition The Systole of a compact metric space X is the least length of a non-contractible loop in X, denoted by sys π 1 . Question Relation of systole with global geometric quantities, which are independent of curvature. 1
Definition d : X × X → R + is called a metric if • d ( x , y ) = d ( y , x ); • d ( x , y ) ≥ 0, and = occurs if and only if x = y ; • d ( x , z ) + d ( z , y ) ≥ d ( x , y ), ∀ x , y , z ∈ X . ( X , d ) is called a metric space. 2
Definition d : X × X → R + is called a metric if • d ( x , y ) = d ( y , x ); • d ( x , y ) ≥ 0, and = occurs if and only if x = y ; • d ( x , z ) + d ( z , y ) ≥ d ( x , y ), ∀ x , y , z ∈ X . ( X , d ) is called a metric space. It is called compact if every sequence in X has a convergent subsequence. 2
A non-contractible loop 3
History 4
History • 1947: Tutte studied the girth of a graph. 4
History • 1947: Tutte studied the girth of a graph. • 1949: Loewner initiated the study on the torus. 4
History • 1947: Tutte studied the girth of a graph. • 1949: Loewner initiated the study on the torus. • 1950’s thesis: P.M. Pu on the projective plane and the M¨ obius band. 4
History • 1947: Tutte studied the girth of a graph. • 1949: Loewner initiated the study on the torus. • 1950’s thesis: P.M. Pu on the projective plane and the M¨ obius band. • 1960-1970’s: Berger’s propaganda. (An impetus from Thom) 4
History • 1947: Tutte studied the girth of a graph. • 1949: Loewner initiated the study on the torus. • 1950’s thesis: P.M. Pu on the projective plane and the M¨ obius band. • 1960-1970’s: Berger’s propaganda. (An impetus from Thom) • 1983: Gromov’s spectacular inequality. 4
History • 1947: Tutte studied the girth of a graph. • 1949: Loewner initiated the study on the torus. • 1950’s thesis: P.M. Pu on the projective plane and the M¨ obius band. • 1960-1970’s: Berger’s propaganda. (An impetus from Thom) • 1983: Gromov’s spectacular inequality. • Higher genus surface, k -systole, filling area, volume entropy, Lusternik-Schnirelmann category, J-holomorphic curves... 4
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Loewner Theorem 2 sys π 1 ( T 2 ) 2 ≤ area ( T 2 ) √ 3 holds for any metric on T 2 . 6
Loewner Theorem 2 sys π 1 ( T 2 ) 2 ≤ area ( T 2 ) √ 3 holds for any metric on T 2 . 2 • 3 = γ 2 , which is the Hermite constant. (lattice) √ 6
Loewner Theorem 2 sys π 1 ( T 2 ) 2 ≤ area ( T 2 ) √ 3 holds for any metric on T 2 . 2 • 3 = γ 2 , which is the Hermite constant. (lattice) √ • Equality holds if T 2 = R 2 / L , where L is the lattice Z -spanned by cubic roots. Hexagonal lattice. 6
Uniformization of Torus ( T 2 , g ) is conformal to a flat torus ( T 2 , g 0 ) i.e., g = ρ 2 g 0 , and any flat tours ( T 2 , g 0 ) is isometric to a parallelogram with two opposite sides identified. This means ( T 2 , g 0 ) is foliated by a family of closed geodesic l s of fixed length l for s ∈ [0 , h ]. By Fubini’s theorem, we have � h � � Area (( T 2 , g )) = T 2 ρ 2 dv 0 = ρ 2 d θ. ds 0 l s 7
H¨ older’s inequality implies: � � � ρ 2 d θ ρ d θ ) 2 . 1 d θ ≥ ( l s l s l s Hence ρ 2 d θ ≥ 1 � l Length g2 ( l s ) . l s Scaling of flat is still flat, we can assume Area (( T 2 , g )) = Area (( T 2 , g 0 )) = h · l . Using mean value theorem, it follows that there exists s 0 such that l ≥ Length g ( l s 0 ) . 8
Hermite constant Hence the problem is reduced to flat tori, which in simple term, we ask for the biggest length of the shorter side among all parallelogram of area 1. 9
P.M. Pu Theorem sys π 1 ( RP 2 ) 2 ≤ π 2 area ( RP 2 ) holds for any metric on RP 2 . 10
P.M. Pu Theorem sys π 1 ( RP 2 ) 2 ≤ π 2 area ( RP 2 ) holds for any metric on RP 2 . • Equality holds if RP 2 is the antipodal quotient of round unit sphere, for which the area is 2 π and the length of a closed geodesic is π . 10
Remarks Isoperimetric inequality: 4 π A ≤ l 2 . • Systolic inequality is kind of a converse to isoperimetric inequality. • Isoperimetric inequality is with respect to a specified metric. • Systolic inequality is a uniform inequality for all metrics. • Higher genus surface: sys π 2 1 ≤ C area , optimal C is not obtained. 11
Gromov Theorem There exists a universal constant C n , such that 1 sys π 1 ( T n ) ≤ C n vol ( T n , g ) n holds for any metric g on T n . 12
One approach: topological dimension theory Theorem (Brouwer 1909) There exists no homeomorphism from R n to R m if n � = m. 13
One approach: topological dimension theory Theorem (Brouwer 1909) There exists no homeomorphism from R n to R m if n � = m. If require the map is linear, we have two stronger facts from linear algebra. • There exists no linear surjection from R n to R m if n < m . • There exists no linear injection from R n to R m if n > m . 13
Dropping the linear requirement, we have 14
Dropping the linear requirement, we have • there exists continuous surjection from R n to R m if n < m . (Peano curves) 14
Dropping the linear requirement, we have • there exists continuous surjection from R n to R m if n < m . (Peano curves) • there exists no continuous injection from R n to R m if n > m . 14
Dropping the linear requirement, we have • there exists continuous surjection from R n to R m if n < m . (Peano curves) • there exists no continuous injection from R n to R m if n > m . Smaller dimension can be stretched to cover a higher-dimensional space, but a higher-dimensional space may not be squeezed to fit into a lower-dimensional space. 14
Open covering { U i } an open covering is called • multiplicity of µ : if any point is contained in at most µ open sets. • diameter D : if diam ( U i ) ≤ D . 15
Lebesgue dimension theory 16
Lebesgue dimension theory • Lebesgue constructed an open covering of R n with multiplicity n + 1 and arbitrary small diameter. 16
Lebesgue dimension theory • Lebesgue constructed an open covering of R n with multiplicity n + 1 and arbitrary small diameter. 16
Lebesgue dimension theory • Lebesgue constructed an open covering of R n with multiplicity n + 1 and arbitrary small diameter. • Lebesgue also proved the following: if U i open sets cover the unit cube of diameter 1, then some point of the n − cube must lie in at least n + 1 different U i . 16
Generalizations to Riemannian manifolds 17
Generalizations to Riemannian manifolds Theorem (Guth 2008) If ( M , g ) is an n dimensional Riemannian manifold with volume V , 1 n . then there is an open cover with multiplicity n and diameter C n V 17
Generalizations to Riemannian manifolds Theorem (Guth 2008) If ( M , g ) is an n dimensional Riemannian manifold with volume V , 1 n . then there is an open cover with multiplicity n and diameter C n V Theorem (Gromov) If ( T n , g ) has systole at least 1 and { U i } an open cover of diameter 1 10 , then some point lies in at least n + 1 different sets U i . 17
Generalizations to Riemannian manifolds Theorem (Guth 2008) If ( M , g ) is an n dimensional Riemannian manifold with volume V , 1 n . then there is an open cover with multiplicity n and diameter C n V Theorem (Gromov) If ( T n , g ) has systole at least 1 and { U i } an open cover of diameter 1 10 , then some point lies in at least n + 1 different sets U i . Hence 1 n . sys π 1 ≤ C n V 17
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