Contact Geometry and Manifolds

Let M be a manifold of dimension 2n + 1. A contact structure on M is a distribution ξ ⊂ TM of dimension 2n, such that the defining 1-form α satisfies

α ∧ (dα)ⁿ ≠ 0 —– (1)

A 1-form α satisfying (1) is said to be a contact form on M. Let α be a contact form on M; then there exists a unique vector field R_α on M such that

α(R_α) = 1, ι_Rα dα = 0,

where ι_Rα dα denotes the contraction of dα along R_α. By definition R_α is called the Reeb vector field of the contact form α. A contact manifold is a pair (M, ξ) where M is a 2n + 1-dimensional manifold and ξ is a contact structure. Let (M, ξ) be a contact manifold and fix a defining (contact) form α. Then the 2-form κ = 1/2 dα defines a symplectic form on the contact structure ξ; therefore the pair (ξ, κ) is a symplectic vector bundle over M. A complex structure on ξ is the datum of J ∈ End(ξ) such that J² = −I_ξ.

Let α be a contact form on M, with ξ = ker α and let κ = 1/2 dα. A complex structure J on ξ is said to be κ-calibrated if g_J [x](·, ·) := κ[x](·, J_x ·) is a J_x–Hermitian inner product on ξ_x for any x ∈ M.

The set of κ-calibrated complex structures on ξ will be denoted by C_α(M). If J is a complex structure on ξ = ker α, then we extend it to an endomorphism of TM by setting

J(R_α) = 0.

Note that such a J satisfies

J² =−I + α ⊗ R_α

If J is κ-calibrated, then it induces a Riemannian metric g on M given by

g := g_J + α ⊗ α —– (2)

Furthermore the Nijenhuis tensor of J is defined by

N_J (X, Y) = [JX, JY] − J[X, JY] − J[Y, JX] + J²[X, Y] for any X, Y ∈ TM

A Sasakian structure on a 2n + 1-dimensional manifold M is a pair (α, J), where

• α is a contact form;

• J ∈ C_α(M) satisfies N_J = −dα ⊗ R_α

The triple (M, α, J) is said to be a Sasakian manifold. Let (M, ξ) be a contact manifold. A differential r-form γ on M is said to be basic if

ι_Rα γ = 0, L_Rα γ = 0,

where L denotes the Lie derivative and R_α is the Reeb vector field of an arbitrary contact form defining ξ. We will denote by Λ^r_B(M) the set of basic r-forms on (M, ξ). Note that

dΛ^r_B(M) ⊂ Λ^r+1_B(M)

The cohomology H^•_B(M) of this complex is called the basic cohomology of (M, ξ). If (M, α, J) is a Sasakian manifold, then

J(Λ^r_B(M)) = Λ^r_B(M), where, as usual, the action of J on r-forms is defined by

J_φ(X₁,…, X_r) = φ(JX₁,…, JX_r)

Consequently Λ^r_B(M) ⊗ C splits as

Λ^r_B(M) ⊗ C = ⊕_p+q=r Λ^p,q_J(ξ)

and, according with this gradation, it is possible to define the cohomology groups H^p,q_B(M). The r-forms belonging to Λ^p,q_J(ξ) are said to be of type (p, q) with respect to J. Note that κ = 1/2 dα ∈ Λ^1,1_J(ξ) and it determines a non-vanishing cohomology class in H^1,1_B(M). The Sasakian structure (α, J) also induces a natural connection ∇^ξ on ξ given by

∇^ξ_X Y = (∇_X Y)^ξ if X ∈ ξ

= [R_α, Y] if X = R_α

where the subscript ξ denotes the projection onto ξ. One easily gets

∇^ξ_X J = 0, ∇^ξ_Xg_J = 0, ∇^ξ_X dα = 0, ∇^ξ_X Y − ∇^ξ_Y X = [X,Y]^ξ,

for any X, Y ∈ TM. Consequently we have Hol(∇^ξ) ⊆ U(n).

The basic cohomology class

c^B₁(M) = 1/2π [ρ^T] ∈ H^1,1_B(M)

is called the first basic Chern class of (M, α, J) and, if it vanishes, then (M, α, J) is said to be null-Sasakian.

Furthermore a Sasakian manifold is called α-Einstein if there exist λ, ν ∈ C^∞(M, R) such that

Ric = λg + να ⊗ α, where Ric is the Ricci Tensor.

A submanifold p: L ֒→ M of a 2n + 1-dimensional contact manifold (M, ξ) is said to be Legendrian if :

1) dim_RL = n,

2) p_∗(TL) ⊂ ξ

Observe that, if α is a defining form of the contact structure ξ, then condition 2) is equivalent to say that p^∗(α) = 0. Hence Legendrian submanifolds are the analogue of Lagrangian submanifolds in contact geometry.

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Credit Bubbles. Thought of the Day 90.0

At the macro-economic level of the gross statistics of money and loan supply to the economy, the reserve banking system creates a complex interplay between money, debt, supply and demand for goods, and the general price level. Rather than being constant, as implied by theoretical descriptions, money and loan supplies are constantly changing at a rate dependent on the average loan period, and a complex of details buried in the implementation and regulation of any given banking system.

Since the majority of loans are made for years at a time, the results of these interactions play out over a long enough time scale that gross monetary features of regulatory failure, such as continuous asset price inflation, have come to be regarded as normal, e.g. ”House prices always go up”. The price level however is not only dependent on purely monetary factors, but also on the supply and demand for goods and services, including financial assets such as shares, which requires that estimates of the real price level versus production be used. As a simplification, if constant demand for goods and services is assumed as shown in the table below, then there are two possible causes of price inflation, either the money supply available to purchase the good in question has increased, or the supply of the good has been reduced. Critically, the former is simply a mathematical effect, whilst the latter is a useful signal, providing economic information on relative supply and demand levels that can be used locally by consumers and producers to adapt their behaviour. Purely arbitrary changes in both the money and the loan supply that are induced by the mechanical operation of the banking system fail to provide any economic benefit, and by distorting the actual supply and demand signal can be actively harmful.

Credit bubbles are often explained as a phenomena of irrational demand, and crowd behaviour. However, this explanation ignores the question of why they aren’t throttled by limits on the loan supply? An alternate explanation which can be offered is that their root cause is periodic failures in the regulation of the loan and money supply within the commercial banking system. The introduction of widespread securitized lending allows a rapid increase in the total amount of lending available from the banking system and an accompanying if somewhat smaller growth in the money supply. Channeled predominantly into property lending, the increased availability of money from lending sources, acted to increase house prices creating rational speculation on their increase, and over time a sizeable disruption in the market pricing mechanisms for all goods and services purchased through loans. Monetary statistics of this effect such as the Consumer Price Index (CPI) for example, are however at least partially masked by production deflation from the sizeable productivity increases over decades. Absent any limit on the total amount of credit being supplied, the only practical limit on borrowing is the availability of borrowers and their ability to sustain the capital and interest repayments demanded for their loans.

Owing to the asymmetric nature of long term debt flows there is a tendency for money to become concentrated in the lending centres, which then causes liquidity problems for the rest of the economy. Eventually repayment problems surface, especially if the practice of further borrowing to repay existing loans is allowed, since the underlying mathematical process is exponential. As general insolvency as well as a consequent debt deflation occurs, the money and loan supply contracts as the banking system removes loan capacity from the economy either from loan repayment, or as a result of bank failure. This leads to a domino effect as businesses that have become dependent on continuously rolling over debt fail and trigger further defaults. Monetary expansion and further lending is also constrained by the absence of qualified borrowers, and by the general unwillingness to either lend or borrow that results from the ensuing economic collapse. Further complications, as described by Ben Bernanke and Harold James, can occur when interactions between currencies are considered, in particular in conjunction with gold-based capital regulation, because of the difficulties in establishing the correct ratio of gold for each individual currency and maintaining it, in a system where lending and the associated money supply are continually fluctuating and gold is also being used at a national level for international debt repayments.

The debt to money imbalance created by the widespread, and global, sale of Asset Backed securities may be unique to this particular crisis. Although asset backed security issuance dropped considerably in 2008, as the resale markets were temporarily frozen, current stated policy in several countries, including the USA and the United Kingdom, is to encourage further securitization to assist the recovery of the banking sector. Unfortunately this appears to be succeeding.