Sheaf Cohomology as the Mathematical Tool Necessary to Describe a Conformally Invariant Isomorphism. Twistors and Spinors Theoreticals. Note Quote.

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The geometry of complex space-time in spinor form calculus is described in terms of spin-space formalism, i.e. a complex vector space endowed with a symplectic form and some fundamental isomorphisms. These mathematical properties enable one to raise and lower indices, define the conjugation of spinor fields in Lorentzian or Riemannian four-geometries, translate tensor fields into spinor fields (or the other way around). The standard two-spinor form of the Riemann curvature tensor is then obtained by relying on the (more) familiar tensor properties of the curvature.

Since the whole of twistor theory may be viewed as a holomorphic description of space-time geometry in a conformally invariant framework, the key results of conformal gravity, i.e. C-spaces, Einstein spaces and complex Einstein spaces necessitates a sufficient condition for a space-time to be conformal to a complex Einstein space.

On to twistor spaces, from the point of view of mathematical physics and relativity theory,  this is defined by twistors as α-planes in complexified compactified Minkowski space-time, and as α-surfaces in curved space-time. In the former case, one deals with totally null two-surfaces, in that the complexified Minkowski metric vanishes on any pair of null tangent vectors to the surface. Hence such null tangent vectors have the form λAπA′ , where λA is varying and πA′ is covariantly constant. This definition can be generalized to complex or real Riemannian four-manifolds, provided that the Weyl curvature is anti-self-dual. An alternative definition of twistors in Minkowski space-time is instead based on the vector space of solutions of a differential equation, which involves the symmetrized covariant derivative of an unprimed spinor field. Interestingly, a deep correspondence exists between flat space-time and twistor space. Hence complex space-time points correspond to spheres in the so-called projective twistor space, and this concept is carefully formulated. Sheaf cohomology can then be used as the mathematical tool necessary to describe a conformally invariant isomorphism between the complex vector space of holomorphic solutions of the wave equation on the forward tube of flat space-time, and the complex vector space of complex-analytic functions of three variables. These are arbitrary, in that they are not subject to any differential equation.

The generalization of Penrose’s non-linear graviton combines two-spinor techniques and twistor theory in a way, where it appears necessary to go beyond anti-self-dual space-times, since they are only a particular class of (complex) space-times, and they do not enable one to recover the full physical content of (complex) general relativity. This implies going beyond the original twistor theory, since the three-complex-dimensional space of α-surfaces only exists in anti-self-dual space-times. Roger Penrose defines twistors as charges for massless spin-3/2 fields. Such an approach has been considered since a vanishing Ricci tensor provides the consistency condition for the existence and propagation of massless helicity-3 fields in curved 2 space-time. Moreover, in Minkowski space-time the space of charges for such fields is naturally identified with the corresponding twistor space. The resulting geometric scheme in the presence of curvature is as follows. First, define a twistor for Ricci-flat space-time. Second, characterize the resulting twistor space. Third, reconstruct the original Ricci-flat space-time from such a twistor space. One of the main technical difficulties of the program proposed by Penrose is to obtain a global description of the space of potentials for massless spin-3/2 fields.

Conjuncted: Mispricings Happened in the Past do not Influence the Derivative Price: Black-Scholes (BS) Analysis and Arbitrage-Free Financial Economics. Note Quote.

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It can be shown that the probability (up to a normalization constant) of the trajectory R(·,·) has the form:

P[R(.,.)] ∼ exp[-1/2∑0 dt dt’ dS dS’ R(t, S) K-1(t, S|t’, S’) R(t’, S’)] —– (1)

where the kernel of the operator K is defined as:

K(t, S|t’, S’) = θ (T – t) θ (T – t’)∫0 dτ ds f(τ) θ (t – τ) θ (t’ – τ) e-λ(t + t’ – 2τ) x P (t, S|τ, s)P (t′, S′|τ, s) —– (2)

It is easy to see that the kernel is of order 1/λ and vanishes as λ → ∞. Equation 2, in particular, results in the equality for the correlation function:

⟨R(t, S) R(t′, S′)⟩ = Σ2 · K(t, S|t′, S′) —– (3)

Local Gauge Transformations in Locally Gauge Invariant Relativistic Field Theory

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The question arises of whether local space-time symmetries – arbitrary co-ordinate transformations that leave the explicit form of the equations of motion unaffected – also have an active interpretation. As in the case of local gauge symmetry, it has been argued in the literature that the introduction of a force is required to ‘restore’ local symmetry.  In the case of arbitrary co-ordinate transformations, the force invoked is gravity. Once again, we believe that the arguments (though seductive) are wrong, and that it is important to see why. Kosso’s discussion of arbitrary coordinate transformations is analogous to his argument with respect to local gauge transformations. He writes:

Observing this symmetry requires comparing experimental outcomes between two reference frames that are in variable relative motion, frames that are relatively accelerating or rotating….One can, in principle, observe that this sort of transformation has occurred. … just look out of the window and you can see if you are speeding up or turning with respect to some object that defines a coordinate system in the reference frame of the ground…Now do the experiments to see if the invariance is true. Do the same experiments in the original reference frame that is stationary on the ground, and again in the accelerating reference frame of the train, and see if the physics is the same. One can run the same experiments, with mechanical forces or with light and electromagnetic forces, and observe the results, so the invariance should be observable…But when the experiments are done, the invariance is not directly observed. Spurious forces appear in the accelerating system, objects move spontaneously, light bends, and so on. … The physics is different.

In other words, if we place ourselves at rest first in an inertial reference frame, and then in a non-inertial reference frame, our observations will be distinguishable. For example, in the non-inertial reference frame objects that are seemingly force-free will appear to accelerate, and so we will have to introduce extra, ‘spurious’, forces to account for this accelerated motion. The transformation described by Kosso is clearly not a symmetry transformation. Despite that, his claim appears to be that if we move to General Relativity, this transformation becomes a symmetry transformation. In order to assess this claim, let’s begin by considering Kosso’s experiment from the point of view of classical physics.

Suppose that we describe these observations using Newtonian physics and Maxwell’s equations. We would not be surprised that our descriptions differ depending on the choice of coordinate system: arbitrary coordinate transformations are not symmetries of the Newtonian and Maxwell equations of motion as usually expressed. Nevertheless, we are free to re-write Newtonian and Maxwellian physics in generally covariant form. But notice: the arbitrary coordinate transformations now apply not just to the Newtonian particles and the Maxwellian electromagnetic fields, but also to the metric, and this is necessary for general covariance.

Kosso’s example is given in terms of passive transformations – transformations of the coordinate systems in which we re-coordinatise the fields. In the Kosso experiment, however, we re-coordinatise the matter fields without re-coordinatising the metric field. This is not achieved by a mere coordinate transformation in generally covariant classical theory: a passive arbitrary coordinate transformation induces a re-coordinatisation of not only the matter fields but also the metric. The two states described by Kosso are not related by an arbitrary coordinate transformation in generally covariant classical theory. Further, such a coordinate transformation applied to only the matter and electromagnetic fields is not a symmetry of the equations of Newtonian and Maxwellian physics, regardless of whether those equations are written in generally covariant form.

Suppose that we use General Relativity to describe the above observations. Kosso suggests that in General Relativity the observations made in an inertial reference frame will indeed be related by a symmetry transformation to those made in a non-inertial reference frame. He writes:

The invariance can be restored by revising the physics, by adding a specific dynamical principle. This is why the local symmetry is a dynamical symmetry. We can add to the physics a claim about a specific force that restores the invariance. It is a force that exactly compensates for the local transform. In the case of the general theory of relativity the dynamical principle is the principle of equivalence, and the force is gravity. … With gravity included in the physics and with the windows of the train shuttered, there is no way to tell if the transformation, the acceleration, has taken place. That is, there is now no difference in the outcome of experiments between the transformed and untransformed systems. The force pulling objects to the back of the train could just as well be gravity. Thus the physics, all things including gravity considered, is invariant from one locally transformed frame to the next. The symmetry is restored.

This analysis mixes together the equivalence principle with the meaning of invariance under arbitrary coordinate transformations in a way which seems to us to be confused, with the consequence that the account of local symmetry in General Relativity is mistaken.

Einstein’s field equations are covariant under arbitrary smooth coordinate transformations. However, as with generally covariant Newtonian physics, these symmetry transformations are transformations of the matter fields (such as particles and electromagnetic radiation) combined with transformations of the metric. Kosso’s example, as we have already emphasised, re-coordinatises the matter fields without re-coordinatising the metric field. So, the two states described by Kosso are not related by an arbitrary coordinate transformation even in General Relativity. We can put the point vividly by locating ourselves at the origin of the coordinate system: I will always be able to tell whether the train, myself, and its other contents are all freely falling together, or whether there is a relative acceleration of the other contents relative to the train and me (in which case the other contents would appear to be flung around). This is completely independent of what coordinate system I use – my conclusion is the same regardless of whether I use a coordinate system at rest with respect to the train or one that is accelerating arbitrarily. (This coordinate independence is, of course, the symmetry that Kosso sought in the opening quotation above, but his analysis is mistaken.)

What, then, of the equivalence principle? The Kosso transformation leads to a physically and observationally distinct scenario, and the principle of equivalence is not relevant to the difference between those scenarios. What the principle of equivalence tells us is that the effect in the second scenario, where the contents of the train appear to accelerate to the back of the train, may be due to acceleration of the train in the absence of a gravitational field, or due to the presence of a gravitational field in which the contents of the train are in free fall but the train is not. Mere coordinate transformations cannot be used to bring real physical forces in and out of existence.

It is perhaps worthwhile briefly indicating the analogy between this case and the gauge case. Active arbitrary coordinate transformations in General Relativity involve transformations of both the matter fields and the metric, and they are symmetry transformations having no observable consequences. Coordinate transformations applied to the matter fields alone are no more symmetry transformations in General Relativity than they are in Newtonian physics (whether written in generally covariant form or not). Such transformations do have observational consequences. Analogously, local gauge transformations in locally gauge invariant relativistic field theory are transformations of both the particle fields and the gauge fields, and they are symmetry transformations having no observable consequences. Local phase transformations alone (i.e. local gauge transformations of the matter fields alone) are no more symmetries of this theory than they are of the globally phase invariant theory of free particles. Neither an arbitrary coordinate transformation in General Relativity, nor a local gauge transformation in locally gauge invariant relativistic field theory, can bring forces in and out of existence: no generation of gravitational effects, and no changes to the interference pattern.

Black-Scholes (BS) Analysis and Arbitrage-Free Financial Economics

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The Black-Scholes (BS) analysis of derivative pricing is one of the most beautiful results in financial economics. There are several assumptions in the basis of BS analysis such as the quasi-Brownian character of the underlying price process, constant volatility and, the absence of arbitrage.

let us denote V (t, S) as the price of a derivative at time t condition to the underlying asset price equal to S. We assume that the underlying asset price follows the geometrical Brownian motion,

dS/S = μdt + σdW —– (1)

with some average return μ and the volatility σ. They can be kept constant or be arbitrary functions of S and t. The symbol dW stands for the standard Wiener process. To price the derivative one forms a portfolio which consists of the derivative and ∆ units of the underlying asset so that the price of the portfolio is equal to Π:

Π = V − ∆S —– (2)

The change in the portfolio price during a time step dt can be written as

dΠ = dV − ∆dS = (∂V/∂t + σ2S22V/2∂S2) dt + (∂V/∂S – ∆) dS —– (3)

from of Ito’s lemma. We can now chose the number of the underlying asset units ∆ to be equal to ∂V/∂S to cancel the second term on the right hand side of the last equation. Since, after cancellation, there are no risky contributions (i.e. there is no term proportional to dS) the portfolio is risk-free and hence, in the absence of the arbitrage, its price will grow with the risk-free interest rate r:

dΠ = rΠdt —– (4)

or, in other words, the price of the derivative V(t,S) shall obey the Black-Scholes equation:

(∂V/∂t + σ2S22V/2∂S2) dt + rS∂V/∂S – rV = 0 —– (5)

In what follows we use this equation in the following operator form:

LBSV = 0, LBS = ∂/∂t + σ2S22V/2∂S2 + rS∂/∂S – r —– (6)

To formulate the model we return back to Eqn(1). Let us imagine that at some moment of time τ < t a fluctuation of the return (an arbitrage opportunity) appeared in the market. It happened when the price of the underlying stock was S′ ≡ S(τ). We then denote this instantaneous arbitrage return as ν(τ, S′). Arbitragers would react to this circumstance and act in such a way that the arbitrage gradually disappears and the market returns to its equilibrium state, i.e. the absence of the arbitrage. For small enough fluctuations it is natural to assume that the arbitrage return R (in absence of other fluctuations) evolves according to the following equation:

dR/dt = −λR,   R(τ) = ν(τ,S′) —– (7)

with some parameter λ which is characteristic for the market. This parameter can be either estimated from a microscopic theory or can be found from the market using an analogue of the fluctuation-dissipation theorem. The fluctuation-dissipation theorem states that the linear response of a given system to an external perturbation is expressed in terms of fluctuation properties of the system in thermal equilibrium. This theorem may be represented by a stochastic equation describing the fluctuation, which is a generalization of the familiar Langevin equation in the classical theory of Brownian motion. In the last case the parameter λ can be estimated from the market data as

λ = -1/(t -t’) log [〈LBSV/(V – S∂V/∂S) (t) LBSV/(V – S∂V/∂S) (t’)〉market / 〈(LBSV/(V – S∂V/∂S)2 (t)〉market] —– (8)

and may well be a function of time and the price of the underlying asset. We consider λ as a constant to get simple analytical formulas for derivative prices. The generalization to the case of time-dependent parameters is straightforward.

The solution of Equation 7 gives us R(t,S) = ν(τ,S)e−λ(t−τ) which, after summing over all possible fluctuations with the corresponding frequencies, leads us to the following expression for the arbitrage return at time t:

R (t, S) = ∫0t dτ ∫0 dS’ P(t, S|τ, S’) e−λ(t−τ) ν (τ, S’), t < T —– (9)

where T is the expiration date for the derivative contract started at time t = 0 and the function P (t, S|τ, S′) is the conditional probability for the underlying price. To specify the stochastic process ν(t,S) we assume that the fluctuations at different times and underlying prices are independent and form the white noise with a variance Σ2 · f (t):

⟨ν(t, S)⟩ = 0 , ⟨ν(t, S) ν (t′, S′)⟩ = Σ2 · θ(T − t) f(t) δ(t − t′) δ(S − S′) —– (10)

The function f(t) is introduced here to smooth out the transition to the zero virtual arbitrage at the expiration date. The quantity Σ2 · f (t) can be estimated from the market data as:

∑2/2λ· f (t) = 〈(LBSV/(V – S∂V/∂S)) 2 (t)⟩ market —– (11)

and has to vanish as time tends to the expiration date. Since we introduced the stochastic arbitrage return R(t, S), equation 4 has to be substituted with the following equation:

dΠ = [r + R(t, S)]Πdt, which can be rewritten as

LBSV = R (t, S) V – (S∂V/∂S) —– (12)

using the operator LBS. 

It is worth noting that the model reduces to the pure BS analysis in the case of infinitely fast market reaction, i.e. λ → ∞. It also returns to the BS model when there are no arbitrage opportunities at all, i.e. when Σ = 0. In the presence of the random arbitrage fluctuations R(t, S), the only objects which can be calculated are the average value and other higher moments of the derivative price.

Yield Curve Dynamics or Fluctuating Multi-Factor Rate Curves

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The actual dynamics (as opposed to the risk-neutral dynamics) of the forward rate curve cannot be reduced to that of the short rate: the statistical evidence points out to the necessity of taking into account more degrees of freedom in order to represent in an adequate fashion the complicated deformations of the term structure. In particular, the imperfect correlation between maturities and the rich variety of term structure deformations shows that a one factor model is too rigid to describe yield curve dynamics.

Furthermore, in practice the value of the short rate is either fixed or at least strongly influenced by an authority exterior to the market (the central banks), through a mechanism different in nature from that which determines rates of higher maturities which are negotiated on the market. The short rate can therefore be viewed as an exogenous stochastic input which then gives rise to a deformation of the term structure as the market adjusts to its variations.

Traditional term structure models define – implicitly or explicitly – the random motion of an infinite number of forward rates as diffusions driven by a finite number of independent Brownian motions. This choice may appear surprising, since it introduces a lot of constraints on the type of evolution one can ascribe to each point of the forward rate curve and greatly reduces the dimensionality i.e. the number of degrees of freedom of the model, such that the resulting model is not able to reproduce any more the complex dynamics of the term structure. Multifactor models are usually justified by refering to the results of principal component analysis of term structure fluctuations. However, one should note that the quantities of interest when dealing with the term structure of interest rates are not the first two moments of the forward rates but typically involve expectations of non-linear functions of the forward rate curve: caps and floors are typical examples from this point of view. Hence, although a multifactor model might explain the variance of the forward rate itself, the same model may not be able to explain correctly the variability of portfolio positions involving non-linear combinations of the same forward rates. In other words, a principal component whose associated eigenvalue is small may have a non-negligible effect on the fluctuations of a non-linear function of forward rates. This question is especially relevant when calculating quantiles and Value-at-Risk measures.

In a multifactor model with k sources of randomness, one can use any k + 1 instruments to hedge a given risky payoff. However, this is not what traders do in real markets: a given interest-rate contingent payoff is hedged with bonds of the same maturity. These practices reflect the existence of a risk specific to instruments of a given maturity. The representation of a maturity-specific risk means that, in a continuous-maturity limit, one must also allow the number of sources of randomness to grow with the number of maturities; otherwise one loses the localization in maturity of the source of randomness in the model.

An important ingredient for the tractability of a model is its Markovian character. Non-Markov processes are difficult to simulate and even harder to manipulate analytically. Of course, any process can be transformed into a Markov process if it is imbedded into a space of sufficiently high dimension; this amounts to injecting a sufficient number of “state variables” into the model. These state variables may or may not be observable quantities; for example one such state variable may be the short rate itself but another one could be an economic variable whose value is not deducible from knowledge of the forward rate curve. If the state variables are not directly observed, they are obtainable in principle from the observed interest rates by a filtering process. Nevertheless the presence of unobserved state variables makes the model more difficult to handle both in terms of interpretation and statistical estimation. This drawback has motivated the development of so-called affine curve models models where one imposes that the state variables be affine functions of the observed yield curve. While the affine hypothesis is not necessarily realistic from an empirical point of view, it has the property of directly relating state variables to the observed term structure.

Another feature of term structure movements is that, as a curve, the forward rate curve displays a continuous deformation: configurations of the forward rate curve at dates not too far from each other tend to be similar. Most applications require the yield curve to have some degree of smoothness e.g. differentiability with respect to the maturity. This is not only a purely mathematical requirement but is reflected in market practices of hedging and arbitrage on fixed income instruments. Market practitioners tend to hedge an interest rate risk of a given maturity with instruments of the same maturity or close to it. This important observation means that the maturity is not simply a way of indexing the family of forward rates: market operators expect forward rates whose maturities are close to behave similarly. Moreover, the model should account for the observation that the volatility term structure displays a hump but that multiple humps are never observed.

What Drives Investment? Or How Responsible is Kelly’s Optimum Investment Fraction?

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A reasonable way to describe assets price variations (on a given time-scale) is to assume them to be multiplicative random walks with log-normal step. This comes from the assumption that growth rates of prices are more significant than their absolute variations. So, we describe the price of a financial assets as a time-dependent multiplicative random process. We introduce a set of N Gaussian random variables xi(t) depending on a time parameter t. By this set, we define N independent multiplicative Gaussian random walks, whose assigned discrete time evolution is given by

pi(t+1) = exi(t)pi(t) —– (1)

for i = 1,…,N, where each xi(t) is not correlated in time. To optimize an investment, one can choose different risk-return strategies. Here, by optimization we will mean the maximization of the typical capital growth rate of a portfolio. A capital W(t), invested into different financial assets who behave as multiplicative random walks, grows almost certainly at an exponential rate ⟨ln W (t+1)/W (t)⟩, where one must average over the distribution of the single multiplicative step. We assume that an investment is diversified according to the Kelly’s optimum investment fraction, in order to maximize the typical capital growth rate over N assets with identical average return α = ⟨exi(t)⟩ − 1 and squared volatility ∆ = ⟨e2xi(t)⟩ − ⟨exi(t)⟩2. It should be noted that Kelly capital growth criterion, which maximizes the expected log of final wealth, provides the strategy that maximizes long run wealth growth asymptotically for repeated investments over time. However, one drawback is found in its very risky behavior due to the log’s essentially zero risk aversion; consequently it tends to suggest large concentrated investments or bets that can lead to high volatility in the short-term. Many investors, hedge funds, and sports bettors use the criterion and its seminal application is to a long sequence of favorable investment situations. On each asset, the investor will allocate a fraction fi of his capital, according to the return expected from that asset. The time evolution of the total capital is ruled by the following multiplicative process

W(t+1) = [1 + ∑i=1Nfi(exi(t) -1)] W(t) —– (2)

First, we consider the case of an unlimited investment, i.e. we put no restriction tothe value of ∑i=1Nfi. The typical growth rate

Vtyp = ⟨ln[1+  ∑i=1Nfi(exi -1)]⟩ —– (3)

of the investor’s capital can be calculated through the following 2nd-order expansion in exi -1, if we assume that fluctuations of prices are small and uncorrelated, that seems to be quite reasonable

Vtyp ≅ ∑i=1Nfi(⟨exi⟩ – 1) – fi2/2(⟨e2xi⟩ – 2⟨exi⟩ + 1 —– (4)

By solving d/df(Vtyp = 0), it easy to show that the optimal value for fi is fiopt (α, Δ) = α / (α2 + Δ) ∀ i. We assume that the investor has little ignorance about the real value of α, that we represent by a Gaussian fluctuation around the real value of α. In the investor’s mind, each asset is different, because of this fluctuation αi = α + εi. The εi are drawn from the same distribution, with ⟨εi⟩ = 0 as errors are normally distributed around the real value. We suppose that the investor makes an effort E to investigate and get information about the statistical parameters of the N assets upon which he will spread his capital. So, his ignorance (i.e. the width of the distribution of the εi) about the real value of αi will be a decreasing function of the effort “per asset” E ; more, we suppose that an even infinite effort will not make N this ignorance vanish. In order to plug these assumptions in the model, we write the width of the distribution of ε as

⟨ε2i⟩ = D0 + (N/E)γ —– (5)

with γ > 0. As one can see, the greater is E, the more exact is the perception, and better is the investment. D0 is the asymptotic ignorance. All the invested fraction fopt (αi, Δ) will be different, according to the investor’s perception. Assuming that the εi are small, we expand all fi(α + εi) in equation 4 up to the 2nd order in εi, and after averaging over the distribution of εi, we obtain the mean value of the typical capital growth rate for an investor who provides a given effort E:

Vtyp = N[A − (D0 + (N/E)γ )B] —– (6)

where

A = (α (3Δ – α2))/(α2 + Δ)3 B = -(α2 – Δ)2/2(α2 + Δ)3 —– (7)

We are now able to find the optimal number of assets to be included in the portfolio (i.e., for which the investment is more advantageous, taken into account the effort provided to get information), by solving d/dNVtyp = 0, it is easy to see that the number of optimal assets is given by

Nopt(E) = E {[(A – D0]/(1 + γ)B}1/γ —– (8)

that is an increasing function of the effort E. If the investor has no limit in the total capital fraction invested in the portfolio (so that it can be greater than 1, i.e. the investor can invest more money than he has, borrowing it from an external source), the capital can take negative values, if the assets included in the portfolio encounter a simultaneous negative step. So, if the total investment fraction is greater than 1, we should take into account also the cost of refunding loss to the bank, to predict the typical growth rate of the capital.

Conjuncted: Philosophizing Twistors via Fibration. Note Quote.

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The fibration is not holomorphic (even when this makes sense) but the fibres are complex submanifolds of the twistor space. Let us now see how we might build such fibrations over more general Riemannian manifolds.

So let N be a 2n-dimensional Riemannian manifold. We may at least construct such a fibration of an almost complex manifold over N as follows: let π:J(N) → N be the bundle of almost Hermitian structures of N. Thus the fibre at x ∈ N is

Jx(N) = {j ∈ End(TxN): j2 = −1, j skew-symmetric}.

This bundle is associated to the orthonormal frame bundle of N with typical fibre J(R2n) = O(2n)/U(n) which is a Hermitian symmetric space (in fact it is two disjoint copies of the compact irreducible Hermitian symmetric space SO(2n)/U(n)). In particular, the typical fibre has an O(2n)-invariant complex structure and thus the vertical distribution V = ker dπ inherits an almost complex structure JV. The Levi-Civita connection on the orthonormal frame bundle induces a horizontal distribution H on J(N) so that we have a splitting

T J (N ) = V ⊕ H

with dπ giving an isomorphism between H and π−1TN. This enables us to define a tautological almost complex structure JH on H by

JjH = j
and adding this to JV gives us an almost complex structure J = JV ⊕ JH on J(N). By

construction, the fibres of π are almost complex submanifolds with respect to J.

If we make a conformal change of metric on N, the bundle J(N) remains unchanged although the horizontal distribution H will vary. However, despite this, it can be shown that the almost complex structure J is independent of the choice of metric within a conformal class. Thus our construction may be viewed as one in Conformal Geometry.

Having got our almost complex structure, it is natural to ask whether or not it is integrable so that J(N) is an honest complex manifold. For this, of course, it is necessary and sufficient that the Nijenhuis tensor NJ of J vanish. The obstruction to this vanishing lies in the curvature tensor of N [22] :

Theorem: Let j ∈ J(N) with √−1-eigenspace T+ ⊂ Tπ(j)NC. Let R denote the Riemann curvature tensor of N. Then NJ vanishes at j if and only if

R(T+, T+)T+ ⊂ T+

Thus J is integrable if the above equation holds for all maximally isotropic subspaces T+ of TNC. This is a condition on the curvature tensor that can be analysed in terms of the representation theory of O(2n) on the space of curvature tensors and one concludes:

Corollary: J is integrable if and only if the Weyl tensor of R vanishes identically (i.e. N is locally conformally flat).

Thus J(N) is a complex manifold only in extremely restricted circumstances. The moral to be drawn from this is that J(N) is “too big” in general for J to be integrable. It is therefore appropriate to seek subbundles of J(N) picked out by the geometry of N in the hope that some of these are complex manifolds. One way to do this is is to restrict attention to those elements of J(N) that are compatible with the holonomy of N.

Figure-6-Holonomy-along-a-leafwise-path