Cotangent Prolongation in Geometric Structures

• Cotangent prolongation is a method that transfers geometric data from a manifold to cotangent or projectivized bundles, underpinning canonical contact, Engel, Norden, and Poisson structures.
• It employs techniques such as classical contact-element models, Cartan prolongation, and spectral sequences to classify invariants like twisting numbers and cohomology classes.
• The construction bridges differential and Poisson geometry by interpreting cotangent paths as coisotropic constraints, leading to integrations via structures like the Weinstein groupoid.

Searching arXiv for the cited papers and closely related usage of “cotangent prolongation” to ground the article. arxiv_search({"query":"id:(Klukas et al., 2011) OR id:(Nannicini, 2018) OR id:(Laurent-Gengoux et al., 2015) cotangent prolongation", "max_results": 10, "sort_by": "relevance"}) arxiv_search({"query":"(Klukas et al., 2011, Nannicini, 2018, Laurent-Gengoux et al., 2015)", "max_results": 10, "sort_by": "relevance"}) I’m going to verify the paper metadata via arXiv search. Cotangent prolongation designates several adjacent but distinct constructions in which geometric data on a manifold is transferred to a cotangent bundle, a projectivized cotangent or contact-element space, or an $S^1$-bundle built from a contact distribution. In one line of work, the classical spaces $S(T^*M)$ and $P(T^*M)$ provide canonical contact manifolds, and the projectivization $P(\xi)$ of a contact plane field $\xi$ carries a canonical Engel distribution. In another, a complex Norden structure $(J,g)$ on $M$ is transported to $T^*M$ through a generalized-geometry isomorphism $\Phi_V$. A related cotangent-path construction in Poisson geometry treats paths in $T^*M$ as a coisotropic constraint surface in a symplectic path space. The common theme is prolongation from base geometry to a cotangent-associated total space, but the resulting structures—contact, Engel, Norden, or Poisson—are not interchangeable (Klukas et al., 2011, Nannicini, 2018, Laurent-Gengoux et al., 2015).

1. Classical contact-element models

A $S(T^*M)$0-dimensional contact manifold is a pair $S(T^*M)$1 where $S(T^*M)$2 is a $S(T^*M)$3-form such that $S(T^*M)$4 everywhere. The hyperplane field $S(T^*M)$5 is maximally non-integrable and is the contact distribution. On the cotangent bundle $S(T^*M)$6 with projection $S(T^*M)$7, the canonical Liouville $S(T^*M)$8-form $S(T^*M)$9 is defined by $P(T^*M)$0; in local coordinates $P(T^*M)$1, $P(T^*M)$2. The unit cotangent bundle $P(T^*M)$3, with respect to a chosen Riemannian metric on $P(T^*M)$4, carries a natural contact form $P(T^*M)$5, and the contact distribution is $P(T^*M)$6. Equivalently, the projectivized cotangent bundle $P(T^*M)$7 has a canonical contact structure induced by the kernel of $P(T^*M)$8 modulo the fiberwise scaling action; the contact distribution is well-defined because $P(T^*M)$9 is homogeneous of degree $P(\xi)$0 under fiberwise dilations (Klukas et al., 2011).

These are the classical spaces of contact elements. They model the contact geometry of directions in $P(\xi)$1, and they supply the background for the prolongation of a contact structure $P(\xi)$2. The projectivized contact-plane bundle $P(\xi)$3 is analogous to $P(\xi)$4, but it projectivizes directions inside $P(\xi)$5 rather than all covector directions in $P(\xi)$6. In this sense, cotangent prolongation in the contact-element paradigm is a restriction from all cotangent directions to the directions selected by the contact distribution.

2. Cartan prolongation and Engel geometry

For a contact $P(\xi)$7-manifold $P(\xi)$8, the Cartan prolongation $P(\xi)$9 is the $\xi$0-bundle over $\xi$1 obtained by projectivizing the contact planes: over $\xi$2, the fiber $\xi$3 is the projective line of $\xi$4, namely the set of lines $\xi$5. An Engel structure on a $\xi$6-manifold $\xi$7 is a rank-$\xi$8 distribution $\xi$9 such that

$(J,g)$0

and its characteristic line field $(J,g)$1 is defined by $(J,g)$2. For the prolongation $(J,g)$3, there is a canonical Engel distribution $(J,g)$4: at a point $(J,g)$5 with $(J,g)$6 a line, one sets

$(J,g)$7

that is, the tangent space of $(J,g)$8 at $(J,g)$9 along the tautological line $M$0 inside the fiber $M$1 (Klukas et al., 2011).

In a local contact framing of $M$2 by vector fields $M$3, one may parametrize $M$4 by an angle $M$5 with

$M$6

Then a local frame for the Engel distribution is

$M$7

and its derived distribution is

$M$8

which has rank $M$9, with characteristic line field $T^*M$0 tangent to the $T^*M$1-fibers. This construction is central in Engel theory because it turns a contact $T^*M$2-manifold into a $T^*M$3-manifold equipped with a canonical maximally non-integrable rank-$T^*M$4 distribution. The fiberwise angular direction is not ancillary; it is exactly the characteristic direction of the prolonged Engel structure.

3. $T^*M$5-fold prolongations, spectral sequences, and obstruction theory

Following Adachi and as refined in the classification results, an $T^*M$6-fold prolongation of $T^*M$7 is an Engel manifold $T^*M$8 where $T^*M$9 is an $\Phi_V$0-bundle over $\Phi_V$1 equipped with an Engel structure $\Phi_V$2 whose characteristic line field is tangent to the $\Phi_V$3-fibers, together with a development map

$\Phi_V$4

that, fiberwise, is the connected $\Phi_V$5-fold covering of $\Phi_V$6. Equivalently, $\Phi_V$7 is a fiberwise $\Phi_V$8-fold covering of $\Phi_V$9-bundles $T^*M$0. The twisting number $T^*M$1 is the degree of $T^*M$2 restricted to each $T^*M$3-fiber; it measures how many times the line $T^*M$4 rotates inside $T^*M$5 along a fiber. If $T^*M$6 is a fiberwise $T^*M$7-fold covering of principal $T^*M$8-bundles over $T^*M$9, then their Euler classes satisfy $S(T^*M)$00. Consequently, a necessary condition for the existence of a fiberwise $S(T^*M)$01-fold covering $S(T^*M)$02 is that the mod-$S(T^*M)$03 reduction of $S(T^*M)$04 vanishes (Klukas et al., 2011).

The obstruction theory is encoded by the Serre spectral sequence for an $S(T^*M)$05-bundle $S(T^*M)$06 with coefficients $S(T^*M)$07. Since $S(T^*M)$08 is nonzero only for $S(T^*M)$09, the $S(T^*M)$10-page

$S(T^*M)$11

yields the exact sequence

$S(T^*M)$12

where the transgression $S(T^*M)$13 sends the generator $S(T^*M)$14 to the mod-$S(T^*M)$15 reduction $S(T^*M)$16 of the Euler class. Thus

$S(T^*M)$17

For a contact manifold $S(T^*M)$18 with prolongation $S(T^*M)$19, an $S(T^*M)$20-fold prolongation exists if and only if

$S(T^*M)$21

When this obstruction vanishes, the isomorphism classes of fiberwise $S(T^*M)$22-fold coverings $S(T^*M)$23 are in one-to-one correspondence with $S(T^*M)$24. Equivalently, Engel $S(T^*M)$25-fold prolongations are classified topologically by $S(T^*M)$26 together with $S(T^*M)$27 and $S(T^*M)$28. A further computation gives

$S(T^*M)$29

where $S(T^*M)$30 is the Gauss map of the oriented plane field $S(T^*M)$31 and $S(T^*M)$32 is the positive generator. It follows that $S(T^*M)$33 reduces to $S(T^*M)$34 modulo $S(T^*M)$35 and modulo $S(T^*M)$36, so every contact $S(T^*M)$37-manifold admits $S(T^*M)$38-fold and $S(T^*M)$39-fold prolongations, whereas for $S(T^*M)$40 obstructions can occur.

4. Topological classification and the additional $S(T^*M)$41 invariant

Let $S(T^*M)$42 be an oriented $S(T^*M)$43-bundle and $S(T^*M)$44 an Engel structure on $S(T^*M)$45 whose characteristic line field $S(T^*M)$46 is tangent to the $S(T^*M)$47-fibers. Then $S(T^*M)$48 determines three invariants: a contact structure $S(T^*M)$49 on $S(T^*M)$50 given by $S(T^*M)$51; a twisting number $S(T^*M)$52, equal to the fiberwise degree of the development map $S(T^*M)$53; and a cohomology class $S(T^*M)$54 that records the monodromy of the fiberwise covering. The classification statement is that $S(T^*M)$55 is classified up to Engel diffeomorphism by the triple $S(T^*M)$56. In particular, fixing only the induced contact structure on a cross-section and the twisting along the $S(T^*M)$57-fibers is not sufficient; one must also fix the class $S(T^*M)$58 (Klukas et al., 2011).

The class $S(T^*M)$59 can be computed concretely. For a fiberwise $S(T^*M)$60-fold covering $S(T^*M)$61 and a loop $S(T^*M)$62 representing a generator in $S(T^*M)$63, one lifts $S(T^*M)$64 to a loop $S(T^*M)$65 in $S(T^*M)$66 and projects to the fiber angle via $S(T^*M)$67; then

$S(T^*M)$68

Extending linearly gives a homomorphism $S(T^*M)$69, hence a class in $S(T^*M)$70. Different choices of $S(T^*M)$71 yield non-equivalent fiberwise coverings, and therefore non-equivalent $S(T^*M)$72-fold prolongations. This refines earlier claims in the literature that only the induced contact structure and twisting number were needed.

The $S(T^*M)$73-torus and $S(T^*M)$74-sphere show the range of this invariant. On $S(T^*M)$75, let

$S(T^*M)$76

with contact framing

$S(T^*M)$77

Then $S(T^*M)$78 is diffeomorphic to $S(T^*M)$79, and

$S(T^*M)$80

Fiberwise $S(T^*M)$81-fold coverings of $S(T^*M)$82 are classified by $S(T^*M)$83. For $S(T^*M)$84, the map

$S(T^*M)$85

produces Engel structures $S(T^*M)$86 on $S(T^*M)$87 that are pairwise non-equivalent through Engel isotopies preserving $S(T^*M)$88 tangent to the $S(T^*M)$89-fibers. By contrast, on $S(T^*M)$90, $S(T^*M)$91 and $S(T^*M)$92, so every $S(T^*M)$93-fold prolongation exists and is unique up to isomorphism.

5. Norden structures on cotangent bundles

A Norden manifold is an almost complex manifold $S(T^*M)$94 equipped with a pseudo-Riemannian metric $S(T^*M)$95 of neutral signature $S(T^*M)$96 such that $S(T^*M)$97 and

$S(T^*M)$98

for all vector fields $S(T^*M)$99. Equivalently, $P(T^*M)$00 is $P(T^*M)$01-symmetric: $P(T^*M)$02 A complex Norden manifold is a Norden manifold whose almost complex structure $P(T^*M)$03 is integrable, i.e. its Nijenhuis tensor vanishes. A Kähler Norden manifold is a Norden manifold for which the Levi–Civita connection $P(T^*M)$04 of $P(T^*M)$05 preserves $P(T^*M)$06, so $P(T^*M)$07. For a complex Norden manifold $P(T^*M)$08, there exists a unique linear connection $P(T^*M)$09 with torsion $P(T^*M)$10 such that

$P(T^*M)$11

$P(T^*M)$12

and

$P(T^*M)$13

Equivalently,

$P(T^*M)$14

so that $P(T^*M)$15. This $P(T^*M)$16 is the natural canonical connection (Nannicini, 2018).

The prolongation mechanism uses generalized geometry. Let $P(T^*M)$17. It carries the canonical symplectic pairing

$P(T^*M)$18

and the canonical neutral metric

$P(T^*M)$19

A linear connection $P(T^*M)$20 on $P(T^*M)$21 induces a bracket

$P(T^*M)$22

which is a Lie bracket iff $P(T^*M)$23 is flat. The same connection defines a splitting

$P(T^*M)$24

and in local coordinates $P(T^*M)$25 the adapted frame is

$P(T^*M)$26

There is a fibrewise isomorphism

$P(T^*M)$27

such that $P(T^*M)$28 if and only if $P(T^*M)$29 is torsion-free, and $P(T^*M)$30 intertwines the brackets if and only if $P(T^*M)$31 is flat.

Given $P(T^*M)$32, one defines on $P(T^*M)$33 the symmetric bilinear form

$P(T^*M)$34

and the generalized complex structure

$P(T^*M)$35

Transporting these through $P(T^*M)$36 gives

$P(T^*M)$37

and $P(T^*M)$38 is a Norden manifold. In the horizontal/vertical frame,

$P(T^*M)$39

while

$P(T^*M)$40

The signature of $P(T^*M)$41 is neutral $P(T^*M)$42. If $P(T^*M)$43 is a complex Norden manifold whose natural canonical connection $P(T^*M)$44 is flat, then $P(T^*M)$45 is integrable, so $P(T^*M)$46 is a complex Norden manifold. If $P(T^*M)$47 is Kähler Norden flat, then $P(T^*M)$48 is also Kähler Norden flat. The structural distinction is that integrability on $P(T^*M)$49 is intrinsic and holds for the canonical choice $P(T^*M)$50, while integrability on $P(T^*M)$51 is extrinsic with respect to $P(T^*M)$52 and requires flatness to kill curvature terms. In the flat model $P(T^*M)$53 with the standard complex structure and neutral metric, the prolonged pair $P(T^*M)$54 is explicitly computable and satisfies $P(T^*M)$55 and $P(T^*M)$56.

6. Cotangent paths, coisotropic constraints, and Poisson geometry

A related cotangent construction begins with a bivector field $P(T^*M)$57. The bivector is Poisson if and only if $P(T^*M)$58, equivalently if the bracket

$P(T^*M)$59

satisfies the Jacobi identity. The cotangent bundle $P(T^*M)$60 carries the Liouville $P(T^*M)$61-form $P(T^*M)$62 and the canonical symplectic form $P(T^*M)$63. On the smooth path space

$P(T^*M)$64

a point is written as $P(T^*M)$65, where $P(T^*M)$66 is the base path and $P(T^*M)$67. The path space inherits a weak symplectic form by transgression: $P(T^*M)$68 The bundle map $P(T^*M)$69 is defined by $P(T^*M)$70, and a cotangent path is a pair $P(T^*M)$71 satisfying

$P(T^*M)$72

In coordinates,

$P(T^*M)$73

The corresponding constraint functions are

$P(T^*M)$74

and the constraint surface $P(T^*M)$75 is their common zero locus (Laurent-Gengoux et al., 2015).

In the local-function framework, coisotropy is formulated algebraically: the ideal of local functionals vanishing on $P(T^*M)$76 must be a Poisson ideal. The canonical statement is that a bivector $P(T^*M)$77 is Poisson if and only if the constraint surface $P(T^*M)$78 of cotangent paths is coisotropic in $P(T^*M)$79. When $P(T^*M)$80, the brackets of the smeared constraints close modulo constraints; conversely, coisotropy forces the anomaly term proportional to $P(T^*M)$81 to vanish. The construction avoids Banach manifold technology and is effective in the periodic case, where cotangent loops generally fail to form a Banach submanifold. In the Poisson case, $P(T^*M)$82 carries the canonical Lie algebroid structure with anchor $P(T^*M)$83 and Koszul bracket

$P(T^*M)$84

Algebroid paths are precisely cotangent paths, and the coisotropic reduction $P(T^*M)$85 is the Weinstein groupoid integrating the Lie algebroid $P(T^*M)$86 when integrable. This is adjacent to the contact and Norden usages of cotangent prolongation, but distinct in both its objects and its reduction-theoretic outcome.