Equality cases for condenser capacity inequalities under symmetrization

A N N A L E S

U N I V E R S I T A T I S M A R I A E C U R I E - S K Ł O D O W S K A L U B L I N – P O L O N I A

VOL. LXVI, NO. 1, 2012 SECTIO A 1–5

ANNA BEDNARSKA

The vertical prolongation of the projectable connections

Abstract. We prove that any first order F²Mm1,m2,n1,n2-natural operator transforming projectable general connections on an (m1, m2, n1, n2)-dimen- sional fibred-fibred manifold p = (p, p) : (pY: Y → Y ) → (pM: M → M ) into general connections on the vertical prolongation V Y → M of p : Y → M is the restriction of the (rather well-known) vertical prolongation operator V lifting general connections Γ on a fibred manifold Y → M into VΓ (the vertical prolongation ofΓ) on V Y → M .

The aim of this paper is to describe all F²M_m1,m2,n1,n2-natural oper- ators transforming projectable general connections on an (m₁, m2, n1, n2)- dimensional fibred-fibred manifolds into general connections on the vertical prolongation V Y → M of p : Y → M . The similar problem for the case of fibred manifolds was solving in [7]. In the paper [1], authors described nat- ural operators transforming connections on fibred manifolds Y → M into connections on V Y → M .

A fibred-fibred manifold is a fibred surjective submersion p = (p, p) : (pY : Y → Y ) → (pM: M → M )

between two fibred manifolds p_Y: Y → Y and pM: M → M covering p : Y → M such that the restrictions of p to the fibres are submersions.

Equivalently, the fibred-fibred manifold is a fibred square p = (p, p_Y, pM, p), i.e. a commutative square diagram with arrows being surjective submersions

2000 Mathematics Subject Classification. Primary 58A20, Secondary 58A32.

Key words and phrases. Fibred-fibred manifold, natural operator, projectable connection.

p : Y → M , p_Y : Y → Y , p_M: M → M and p : Y → M such that the system (p, p_Y) : Y → M ×_M Y of maps p and p_Y is a submersion, [2], [6].

If p¹ = (p¹, p¹) : (p¹_Y1: Y¹ → Y¹) → (p¹_M1: M¹ → M¹) is another fibred- fibred manifold, then a fibred-fibred map f : Y → Y¹ is the system f = (f, f₁, f₂, f ) of four maps f : Y → Y¹, f₁: Y → Y¹, f₂: M → M¹ and f : M → M¹ such that the relevant cubic diagram is commutative.

A fibred-fibred manifold p = (p, p) : (p_Y : Y → Y ) → (pM: M → M ) is of the dimension (m₁, m2, n1, n2) if dim Y = m1+ m2+ n1+ n2, dim M = m₁+ m₂, dim Y = m₁+ n₁ and dim M = m₁. All fibred-fibred manifolds of the dimension (m₁, m2, n1, n2) and their all local fibred-fibred diffeomor- phisms form the local admissible category over manifolds in the sense of [3], which we denote by F²M_m1,m2,n1,n2. Any F²M_m1,m2,n1,n2-object is locally isomorphic to the trivial fibred square R^m1,m2,n1,n2 with vertices R^m1 × R^m2 × Rⁿ¹ × Rⁿ², R^m1 × R^m2, R^m1 × Rⁿ¹, R^m1 and arrows being obvious projections.

The vertical functor V : F²M_m1,m2,n1,n2 → F M (on (m₁, m₂, n₁, n₂)- dimensional fibred-fibred manifolds) is the usual vertical functor V : F M → F M on fibred manifolds, where F M is the category of all fibred mani- folds and their morphisms. More precisely, V : F²M_m1,m2,n1,n2 → F M is the functor assigning to any (m₁, m2, n1, n2)-dimensional fibred-fibred manifold p = (p, p) : (p_Y : Y → Y ) → (p_M: M → M ) the vertical bun- dle V Y → Y of the corresponding fibred manifold p : Y → M and to any F²M_m1,m2,n1,n2-map f = (f, f₁, f₂, f ) between (m₁, m₂, n₁, n₂)-dimensional fibred-fibred manifolds p = (p, p) : (p_Y: Y → Y ) → (p_M: M → M ) and p¹ = (p¹, p¹) : (p¹_Y1: Y¹ → Y¹) → (p¹_M1: M¹ → M¹) the vertical prolonga- tion V f : V Y → V Y¹ of the corresponding fibred map f = (f, f₂) between corresponding fibred manifolds p : Y → M and p¹: Y¹ → M¹. Obviously, V : F²M_m1,m2,n1,n2 → F M is a bundle functor in the sense of [3].

A projectable general connection on a fibred-fibred manifold p = (p, p) : (pY: Y → Y ) → (pM: M → M ) is a pair Γ = (Γ, Γ) of general connections Γ : Y ×MT M → T Y and Γ : Y ×MT M → T Y on fibred manifolds p : Y → M and p: Y → M , respectively, such that T p_Y ◦ Γ = Γ ◦ (p_Y ×_pM T p_M), [3], [5].

The vertical prolongation VΓ of a projectable general connection Γ = (Γ, Γ) on an F²M_m1,m2,n1,n2-object p = (p, p) : (p_Y : Y → Y ) → (p_M: M → M ) is the vertical prolongation VΓ on V Y → M of the corresponding general connection Γ on the corresponding fibred manifold p : Y → M . (The vertical prolongation of a general connection Γ on a fibred manifold Y → M is the general connection VΓ on V Y → M as in Section 31.1 in [3]. The vertical prolongation of connections on fibred manifolds was also described in [4].)

The general concept of natural operator can be found in [3]. In partic- ular, an F²M_m1,m2,n1,n2-natural operator D transforming projectable gen- eral connections Γ = (Γ, Γ) on an (m₁, m₂, n₁, n₂)-dimensional fibred-fibred manifold p = (p, p) : (p_Y: Y → Y ) → (pM: M → M ) into general connec- tions D(Γ) on V Y → M is a family of F²M_m1,m2,n1,n2-invariant regular operators

D : Con_proj(Y → M ) → Con(V Y → M )

for all F²M_m1,m2,n1,n2-objects p = (p, p) : (pY: Y → Y ) → (pM: M → M ), where Con_proj(Y → M ) is the set of all projectable general connections on p = (p, p) and Con(V Y → M ) is the set of all general connections on V Y → M . The F²M_m1,m2,n1,n2-invariance means that if Γ ∈ Con_proj(Y → M ) and Γ¹ ∈ Con_proj(Y¹ → M¹) are f -related by F²M_m1,m2,n1,n2-map f : Y → Y¹ (i.e. T f ◦ Γ = Γ¹ ◦ (f × T f₂)), then D(Γ) and D(Γ¹) are V f -related (i.e.

T V f ◦ D(Γ) = D(Γ¹) ◦ (f × T f₂)). The regularity for D means that D transforms smoothly parametrized families of connections into smoothly parametrized families of connections.

Thus (by the canonical character of the vertical prolongation of pro- jectable general connections) the family V of operators

V : Con_proj(Y → M ) → Con(V Y → M ), V(Γ) := VΓ

for all F²M_m1,m2,n1,n2-objects p = (p, p) : (p_Y : Y → Y ) → (pM: M → M ) is an F²M_m1,m2,n1,n2-natural operator.

One can verify that V is the first order operator (it means that if Γ, Γ¹ ∈ Con_proj(Y → M ) have the same 1-jets j_x¹(Γ) = j_x¹(Γ¹) at x ∈ M , then it holds V(Γ) = V(Γ¹) over x).

Theorem 1. The operator V is the unique first order F²M_m1,m2,n1,n2- natural operator transforming projectable general connections on (m₁, m₂, n1, n2)-dimensional fibred-fibred manifolds p = (p, p) : (pY : Y → Y ) → (pM: M → M ) into general connections on V Y → M .

For j = 1, . . . , m₂, s = 1, . . . , n₂ we put [j] := m₁+ j and < s >:= n1+ s.

Let xⁱ, x^[j], y^q, y^<s> be the usual coordinates on the trivial fibred square R^m1,m2,n1,n2.

Lemma 1. Let Γ = (Γ, Γ) be a projectable general connection on an (m₁, m2, n1, n2)-dimensional fibred-fibred manifold p = (p, p) : (pY : Y → Y ) → (pM: M → M ) and let y0 ∈ Y be a point. Then there exists an F²M_m1,m2,n1,n2-chart ψ on Y satisfying conditions ψ(y₀) = (0, 0, 0, 0) and j_(0,0,0,0)¹ ψ∗Γ = j_(0,0,0,0)¹ Γ, wheree

(1) Γ =e

m1

X

i=1

dxⁱ⊗ ∂

∂xⁱ+

m2

X

j=1

dx^[j]⊗ ∂

∂x^[j]

+

m1

X

i1,i2=1 n1

X

q=1

A^q_i

1i2xⁱ¹dxⁱ² ⊗ ∂

∂y^q+

m1

X

i1,i2=1 n2

X

s=1

B_i^s_1i2xⁱ¹dxⁱ² ⊗ ∂

∂y^<s>

+

m1

X

i=1 m2

X

j=1 n2

X

s=1

C_ij^sxⁱdx^[j]⊗ ∂

∂y^<s>+

m1

X

i=1 m2

X

j=1 n2

X

s=1

D^s_jix^[j]dxⁱ⊗ ∂

∂y^<s>

+

m2

X

j1,j2=1 n2

X

s=1

E_j^s_1j2x^[j1]dx^[j2]⊗ ∂

∂y^<s>

for some real numbers A^q_i

1i2, B_i^s

1i2, C_ij^s, D_ji^s and E_j^s

1j2 satisfying (2) A^q_i

1i2 = −A^q_i

2i1, B_i^s_1i2 = −B_i^s_2i1, C_ij^s = −D_ji^s, E_j^s_1j2 = −E_j^s_2j1 for i, i₁, i2= 1, . . . , m1, j, j₁, j2 = 1, . . . , m2, q = 1, . . . , n₁, s = 1, . . . , n₂. Proof. Choose a projectable torsion-free classical linear connection ∇ on p_M: M → M , i.e. a torsion-free classical linear connection ∇ on M such that there exists a unique classical linear connection ∇ on M which is p_M-related with ∇. By Lemma 4.2 [5], there exists an F²M_m1,m2,n1,n2-chart ψ on Y covering a ∇-normal fiber coordinate system on M with the center x₀ = p(y₀) such that ψ(y₀) = (0, 0, 0, 0) and such that j_(0,0,0,0)¹ (ψ∗Γ) = j_(0,0,0,0)¹ eΓ, where eΓ means (1) for some real numbers: A^q_i

1i2, B^s_i

1i2, C_ij^s, D^s_ji and E_j^s

1j2

satisfying the condition (2) for i, i₁, i₂ = 1, . . . , m₁, j, j₁, j₂ = 1, . . . , m₂,

q = 1, . . . , n1, s = 1, . . . , n₂. 

Using Lemma 1, one can prove Theorem 1 as follows.

Proof. Let D be an F²M_m1,m2,n1,n2-natural operator of the first order.

Put ∇(Γ) := D(Γ) − V(Γ) : V Y → T^∗M ⊗ V (V Y ). It is sufficient to prove that it holds ∇(Γ) = 0. Because of Lemma 1, the first order of ∆ and invariance of ∇ with respect to charts of the fibred-fibred manifold, it is sufficient to prove that h∆(Γ)_|v, ui = 0 for any u ∈ T_(0,0)(R^m1 × R^m2), any v ∈ (V R^m1,m2,n1,n2)_(0,0,0,0) and any projectable general connection Γ on R^m1,m2,n1,n2 of the form (1) for any real numbers A^q_i

1i2, B_i^s

1i2, C_ij^s, D^s_ji and E_j^s

1j2 satisfying (2) for i, i₁, i₂ = 1, . . . , m₁, j, j₁, j₂ = 1, . . . , m₂, q = 1, . . . , n1, s = 1, . . . , n₂. Using the invariance of ∆ with respect to the base homotheties (txⁱ, tx^[j], y^q, y^<s>) for t > 0, we obtain the condition of homogeneity of the form

h∆(Γ^t)_|v, ui = th∆(Γ)_|v, ui,

where Γ^t means Γ with the coefficients t²A^q_i1i2, t²B_i^s_1i2, t²C_ij^s, t²D_ji^s, t²E_j^s_1j2 instead of A^q_i

1i2, B^s_i

1i2, C_ij^s, D^s_ji,E_j^s

1j2.

By the homogeneous function theorem this type of homogeneity yields

directly that h∆(Γ)_|v, ui = 0. 

In case of m₁ = m, n1 = n, m2 = 0, n2 = 0 we have F²M_m,0,n,0 = F M_m,n, the category of the (m, n)-dimensional fibred manifolds and their local fibre diffeomorphisms. In this case, the projectable general connections are the general connections.

Corollary 1. The operator V is a unique F M_m,n-natural operator of the first order transforming the general connections on an (m, n)-dimensional fibred manifold p : Y → M into the general connections on V Y → M .

References

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[4] Kol´aˇr, I., Mikulski, W. M., Natural lifting of connections to vertical bundles, The Proceedings of the 19th Winter School “Geometry and Physics” (Srn´ı, 1999). Rend.

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Anna Bednarska Institute of Mathematics

Maria Curie-Skłodowska University pl. Marii Curie-Skłodowskiej 1 20-031 Lublin

Poland

e-mail: bednarska@hektor.umcs.lublin.pl Received June 15, 2011