3
$\begingroup$

Let $M$ be a smooth $n$-manifold. A projective structure on $M$ is a class $p$ of torsion-free connections on $TM$ which have the same geodesics as unparametrized curves.

The bundle of densities of projective weight $w$ is the density bundle $\mathcal{E}(w) \to M$ associated to the (tangent) frame bundle of $TM \to M$ for the representation $g \mapsto |\det(g)|^{\frac{w}{n+1}}$ of $GL(n, \mathbb{R})$ on $\mathbb{R}$. Now the standard projective cotractor bundle is defined to be the jet bundle $$\mathcal{T}^* := J^1 \mathcal{E}(1) \to M.$$

Now from the point of view of Cartan geometry and Klein geometry, the usual flat model for $n$-dimensional projective geometry is the projective sphere (the double cover of projective space) expressed as a homogeneous space $SL(n+1, \mathbb{R})/P$, where $P$ is the stabilizer of the ray through the first standard basis vector in $\mathbb{R}^{n+1}$.

In the Cartan geometric picture of projective geometry, the standard projective cotractor bundle is the associated bundle to the Cartan principal $P$-bundle for the contragradient representation $g \mapsto (g^{-1})^*$ of $P$ on ${\mathbb{R}^{n+1}}^*$. Thus, for the projective cotractor bundle $\mathcal{T}^* = J^1\mathcal{E}(1)$ we should be able to write down a Cech cocycle of transition functions with values in $P$. Is there any obvious way to see how to do this?

$\endgroup$

1 Answer 1

3
$\begingroup$

There are several aspects to your question, and I think that asking for a cocycle of transition functions is partly misleading. The point here is that as you observe in the quesiton, you can define $\mathcal E(1)$ as a density bundle, without making reference to a projective structure. Thus also $J^1\mathcal E(1)$ is, as a vector bundle, independent of the projective structure. Likewise, the canonical Cartan bundle associated to a projective structure is, as a principal fiber bundle, independent of the projective strucutre. (You can define it as $\mathcal PM\times_{GL(n,\mathbb R)}P$, i.e. by an extension of structure group of the linear frame bundle of $M$.) The actual projective structure is not encoded in these bundles but in the canonical linear connection on $\mathcal T^*$ or equivalently the canonical Cartan connection.

To understand $\mathcal T^*$ as a vector bundle, recall that there is a canonical exact sequence $T^*M\otimes\mathcal E(1)\to J^1\mathcal E(1)\to \mathcal E(1)$ (the jet exact sequence). This alrady contains all the information needed to see that it is naturally associated to the representation $\mathbb R^{(n+1)*}$ of $P$. Since $P$ is the stabilizer of a line in $\mathbb R^{n+1}$ it can be equivalently realized as the stabilizer of a hyperplane in $\mathbb R^{(n+1)*}$. (The hyperplane is the annihilator of the distinguished line.) Being associated in this sense just means containing a distinguished subbundle of hyperplanes.

Now you can further recall that choosing a linear connection on $\mathcal E(1)$ defines a splitting of the jet exact sequence, so as a vector bundle $J^1\mathcal E(1)\cong (T^*M\otimes \mathcal E(1))\oplus\mathcal E(1)$, which tells you how to write down transition functions from those of $T^*M$ and $\mathcal E(1)$. For any projective class $[\nabla]$ mapping a connection on $T^*M$ to the induced connection on $\mathcal E(1)$ defines an isomorphism between connections in the projective class and connections on $\mathcal E(1)$, which leads to the standard picture of splittings of $\mathcal T^*$ associated to the connections in the projective class.

The reason why I consider the question of transition functions as partly misleading is that the description of $\mathcal T^*$ as $J^1\mathcal E(1)$ also tells you that diffeomorphisms of $M$ act on $\mathcal T^*$ in a specific way. This information easily gets lost if you describe the bundle in terms of transition functions. The "right" action is given as follows: A diffeomorphism $f:M\to M$ induces a pullback operator $f^*$ on sections of $\mathcal E(1)$, basically given by composing with $f$ and multiplying by an appropriate power of $|det(Df)|$. As you can see, $f^*\sigma(x)$ alredy depends on first derivatives of $f$ in $x$. The natural action on $J^1\mathcal E(1)$ then basically maps $j^1_x\sigma$ to $j^1_x(f^*\sigma)$, so this depends on second derivatives of $f$ in $x$. Hence $\mathcal T^*$ is not associated to the frame bundle of $M$ but only to the second order frame bundle. (Things can be similarly phrased in terms of the Cartan bundle.) With the "right" action of diffeomorphisms, it is then true that a diffeomorphism preserves the projective class (which singles out a finite dimensional subgroup of the diffeomorphism group) if and only if the induced map on $\mathcal T^*$ is compatible with the tractor connection.

$\endgroup$
2
  • $\begingroup$ Thanks! This is really helpful. Do you explain these sort of ideas in any of your papers or in your textbook? I've been having difficulty relating these tractor bundle constructions (in projective and conformal geometry) to the Cartan-geometric picture. $\endgroup$
    – ಠ_ಠ
    Commented Nov 6, 2016 at 21:26
  • 1
    $\begingroup$ If you look at a geometry of type $(G,P)$, then tractor bundles are just associated bundles corresponding to representations of $P$ which are restrictions of representations of $G$. These are easily seen to carry a canonical linear connection. Conversely, the Cartan geometry can be reconstructed as an adapted frame bundle. There are two old articles of Rod Gover and myself (MR1873017 and MR1822358) on this as well as a sketch for the conformal case in MR1996768 . In the textbook these issues are covered in 1.5.7 (Cartan->tractor) and 3.1.19-3.1.22 (tractor->Cartan) in a more general setting. $\endgroup$ Commented Nov 8, 2016 at 8:02

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.