The h-principle for Legendrian immersions

May 17, 2022

Consider a bicycle on a 2-dimensional cartesian plane. We will choose coordinates \((x, y)\) for the position of the bicycle, and angle \(\theta\) that the steering wheel makes with the \(x\)-axis. In other words, our model of the bicycle has three degrees of freedom, two of them given by position in \(\Bbb R^2\) and the other given by the angle of the steering wheel in \(S^1\). Thus, the configuration space of the bicycle is \(M = \Bbb R^2 \times S^1\). The trajectory of such a bicycle in motion \(\gamma(t) = (x(t), y(t))\) on \(\Bbb R^2\) must have tangent \(\gamma'(t)\) parallel to the direction \((\cos \theta(t), \sin \theta(t))\) of the steering wheel. In other words, the motion of the bicycle is constrained by the differential relation \(\dot{x} \cos(\theta) - \dot{y} \sin(\theta) = 0\) on \(M\). The relation can be described as a rank 2 subbundle \(\xi \subset TM\) given by kernel \(\xi = \ker \alpha\) of a differential \(1\)-form \(\alpha = \cos(\theta) dx - \sin(\theta) dy\) on \(M\). Motions of trajectories of the bicycle are therefore paths \(\gamma\) in \(M\) which are parallel to the distribution \(\xi\), i.e., \(\gamma' \in \xi\).

The situation here is the simplest example of a so-called contact manifold. A contact manifold is a manifold \(M\) of odd dimension \(\dim M = 2n+1\) equipped with a \(1\)-form \(\alpha\) which is completely nonintegrable, i.e., \(\alpha \wedge (d\alpha)^{\wedge n}\) is nowhere-vanishing. Contact manifolds are studied upto contactomorphisms, which are diffeomorphisms preserving the conformal class of the contact \(1\)-forms. Therefore, contactomorphisms preserve the associated hyperplane distribution \(\xi = \ker \alpha\), which carries with it a conformal class of a fiberwise symplectic structure given by \(d\alpha\); we denote this class by \(\mathrm{CS}(\xi)\). It is a theorem that any contact manifold is locally contactomorphic to \((\Bbb R^{2n+1}, \xi_0)\) where

\[\xi_0 = \ker(dz - \sum_{i = 1}^n y_i dx_i)\]

This can be seen in the bicycle example, as one can locally scale appropriately so that the form becomes \(dx - \tan(\theta) dy\) and use \(\tan(\theta)\) as a coordinate. Contact manifolds are the odd-dimensional counterparts to symplectic manifolds, which typically occur in physics as phase spaces \(T^*Q\) of a manifold \(Q\). The analogous example in the contact world is the \(1\)-jet space \(J^1 Q = \Bbb R \times T^* Q\), where the contact form is defined by \(dz - \sum_i p_i dq_i\) where \(p_i, q_i\) are the generalized position and momentum coordinates of the phase space. Note that \(\sum_i p_i dq_i\) is the tautological \(1\)-form on \(T^* Q\), and its exterior derivative \(\sum_i dp_i \wedge dq_i\) is exactly the symplectic form on the phase space. Functions \(f : Q \to \Bbb R\) give rise to Lagrangian submanifolds of \(T^* Q\) given by image of the exact \(1\)-form considered as a section \(df : Q \to T^* Q\). Likewise, functions \(f : Q \to \Bbb R\) give rise to certain special submanifolds of \(J^1 Q\) given by image of the \(1\)-jet prolongation \(J^1(f) : Q \to J^1 Q\). These are called Legendrian submanifolds; we give a general definition below.

An immersion \(f : N \to (M^{2n+1}, \xi)\) into a contact manifold is called isotropic if \(f\) is \(\xi\)-parallel, i.e., \(df\) maps \(TN\) to \(\xi\). This imposes the condition that \(\dim N \leq n\), and we say \(f\) is Legendrian if the equality \(\dim N = n\) is achieved. Since \(3 = 2 \cdot 1 + 1\), the trajectories of motion of a bicycle were Legendrian paths in \(\Bbb R^2 \times S^1\). Bicycle tracks have interesting properties: despite being constrained by a closed differential relation (in this case, a differential equation), one can travel from any point to any other point using a bicycle (that’s part of the point of a bicycle, really…). Not only that, but it’s possible to carry out impressive maneuvers to navigate through virtually any set of obstacles. The following theorem might shed some light into this phenomenon. Let us call a pair \((F, f) : (TN, N) \to (TM, M)\) of maps to be a formal Legendrian immersion if \(F\) is a bundle monomorphism covering \(f\) and \(F\) is fiberwise Legendrian (guess what this means). Any Legendrian immersion \(f\) gives rise to a formal Legendrian immersion \((df, f)\), and to distinguish this sub-class of formal immersions we call them holonomic. When we speak of homotopies, we shall think of them as happening inside this “space” of formal Legendrian immersions. We also equip \(M\) with an arbitrary ambient Riemannian metric.

Theorem. Any formal Legendrian immersion is homotopic to a holonomic Legendrian immersion by a \(C^0\)-small homotopy. The result is also true for

CW-complex parametrized families of Legendrian immersions and
relative to submanifolds on which the formal immersions are already holonomic.

We will deduce this from Eliashberg-Mishachev’s holonomic approximation theorem. The form of the theorem we shall need is the following. Let \(X \to V\) be a fiber bundle, \(\mathcal{R} \subset X^{(r)}\) be a subset of the total space of the associated \(r\)-jet bundle, called a differential relation of order \(r\) (for example, the constraint on bicycle trajectories can be viewed as a differential relation of order \(1\)). Sections \(V \to X^{(r)}\) which land inside \(\mathcal{R}\) will be called formal sections of \(\mathcal{R}\), and among these, those which come from \(r\)-jet prolongation of a section \(V \to X\) will be called holonomic sections. We shall introduce two important properties of a differential relation before stating the theorem: Let \(\phi : I^k \to V\) be a continuous map, \(\{f_t : \{\phi(t)\} \to V : t \in I^k\}\) a continuous family of formal sections and \(\{\widetilde{f}_t : \mathrm{Op}(\phi(t)) \to V : t \in \partial I^k\}\) be a family of holonomic extensions of \(f_t\) for \(t \in \partial I^k\). Then we say \(\mathcal{R}\) is (parametrically) locally integrable if there exists a continuous family of holonomic extensions \(\{\widetilde{f}_t : \mathrm{Op}(\phi(t)) \to V : t \in I^k\}\) agreeing with the earlier family for all \(t \in \partial I^k\). For \(k = 0\), note that this simply says every jet in \(\mathcal{R}\) is value of the \(r\)-jet prolongation of some section of \(\mathcal{R}\) at the basepoint. Let \(D^k \times D^{n-k} \cong H\) denote a handle, and let \(C = D^k \times \{0\} \subset H\) denote the core. Let \(\phi : H \times I^m \to V\) be an isotopy, and let \(H_t := \phi(H \times \{t\})\), \(C_t := \phi(C \times \{t\}) \subset H_t\) for all \(t \in I^m\). Suppose we are given (a) a family of germs of holonomic sections \(\{f_t : \mathrm{Op}(H_t) \to \mathcal{R}\}\) near the handles, and (b) a family-homotopy of germs of holonomic sections \(\{f_{t, s} : \mathrm{Op}(\partial H_t \cup C_t) \to \mathcal{R}\}\), \(s \in [0, 1]\) near the union of the boundary and the core of the handles, such that \(f_t\) extends \(f_{t, 0}\), and \(\{f_{t, s}\}\) is a constant in \(s\) on \(\mathrm{Op}(\partial H_t \cap C_t)\). The relation \(\mathcal{R}\) is said to be (parametrically) microflexible if there exists \(\varepsilon > 0\), depending on \(\phi\) and the data given, such that there is a homotopy of germs of holonomic sections \(\widetilde{f}_{t, s} : \mathrm{Op}(H_t) \to \mathcal{R}\), \(s \in [0, \varepsilon]\) extending the given data while staying constant on \(\mathrm{Op}(\partial H_t \cap C_t)\).

Theorem. (Eliashberg-Mishachev) \(\mathcal{R} \subset X^{(r)}\) be a locally integrable microflexible differential relation of order \(r\). Let \(K \subset V\) be a positive codimension subcomplex and \(f : \mathrm{Op}(K) \to \mathcal{R}\) be a germ of a formal section. Then for any arbitrarily small \(\delta, \varepsilon > 0\), there exists a diffeotopy \(\{h_t : V \to V\}\) which is \(\delta\)-small in \(C^0\)-norm, and a holonomic section \(\widetilde{f} : \mathrm{Op}(h_1(K)) \to \mathcal{R}\) such that \(\widetilde{f}, f\) are uniformly \(\varepsilon\)-close on \(\mathrm{Op}(h_1(K)) \subset \mathrm{Op}(K)\).

Suppose \((M, \xi_M), (N, \xi_N)\) are two contact manifolds. An immersion \(f : M \to N\) is called isocontact if \(f^*\mathrm{CS}(\xi_N) = \mathrm{CS}(\xi_M)\). A map from \(M\) to \(N\) is equivalently a section of the product bundle \(X = M \times N\) over \(M\), and the property of being an isocontact immersion can be formulated as a differential relation \(\mathcal{R} \subset X^{(1)}\). \(\mathcal{R}\) is locally integrable by Gray’s stability theorem, as any linear isocontact embedding can be exponentiated to contact charts. \(\mathcal{R}\) is also microflexible, as the contact isotopy extension theorem holds with no restrictions (an interesting remark here is that the same cannot be said for the isosymplectic version, as the symplectic isotopy extension demands \(H^2(H, \partial H \cup C) = 0\) which is true iff \(H\) is not a \(1\)-handle.) Thus, holonomic approximation theorem applies: a formal isocontact immersions to \(N\) near a positive codimension subcomplex of \(M\) can be homotoped to a holonomic isocontact immersion on a \(C^0\)-slightly perturbed domain. In fact, the effect of the perturbation of the domain can be reversed by the contact isotopy extension theorem, using the observation that \(\mathcal{R}\) is invariant under the action of \(\mathrm{Cont}(M)\) on \(X^{(1)} = J^1(M, N)\).

Now suppose \(V\) is an arbitrary manifold, \((M, \xi_M)\) is a contact manifold, and

\[(F, f) : (TV, V) \to (TM, M)\]

is a formal Legendrian immersion. \(F\) maps \(TV\) fiberwise into \((\xi_M, \mathrm{CS}(\xi_M))\), therefore there is a canonical symplectic complement of \(F(TV)\) in \(f^*\xi\), isomorphic to \(T^*V\). Choosing a co-orientation for the distribution \(\xi\), we get a trivial line bundle complementary to \(\xi \subset TM\). Combining, we get that \(F(TV) \subset f^* TM\) has a complementary bundle isomorphic to \(T^*V \times \Bbb R\). This provides an extension of \((F, f)\) to a formal isocontact immersion \((\widehat{F}, \widehat{f}) : (T J^1V, J^1V) \to (TM, M)\). Applying the isocontact holonomic approximation theorem near the zero section as in the last paragraph, we obtain a holonomic Legendrian immersion \(\widetilde{f} : V \to M\) homotopic through formal Legendrian immersions to \((F, f)\) by a \(C^0\)-small homotopy. This finishes the proof.

In particular, this says any immersed path in \(\Bbb R^2\), no matter how complicated, can be approximated by bicycle trajectories. This is because any path in \(\Bbb R^2\) may be lifted to an immersion \(\Bbb R^2 \times S^1\) and upgraded to a formal Legendrian immersion, at which point one can use the result above. The projection of the Legendrian path back to \(\Bbb R^2\) (the “front projection”) need no longer be smooth, and in fact will contain many cusps: imagine the maneuvers needed to park a car in a tight space, and the track left by the process. In an appropriate sense, the holonomic approximation is possible precisely because of these zig-zags through cusps, which enable the tangent vectors to be parallel to the very twisty contact distribution.