We can define projective structure on a manifold in two ways. First we can define it as a maximal atlas of charts from the open subsets of manifold to the projective space, such that transitions maps are locally the elements of the projective general linear group. Second, we can define it as a torsion-free projectively flat connection. Projectively flat connection is a connection which is projectively equivalent with a flat connection around each point of the manifold. Also, two connections are projectively equivalent when there is a closed one-form such that we can write: $D'(X,Y) = D(X,Y) + F(X)Y+F(Y)X$, $D$ and $D'$ are two connections, $X$ and $Y$ are two vector fields and $F$ is our closed one-form. Why we need to define projectively equivalent connections? What is it's interpretation and the relation to the projective space and connections?
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5$\begingroup$ The key point is that a (torsion-free) connection on an $n$-manifold is projectively flat if and only if each point has a neighborhood that possesses a coordinate chart that takes the geodesics of the connection to straight lines in $\mathbb{R}^n$. Two such coordinate charts will have a projective overlap function. The other point is that a connection is projectively flat if and only if it is projectively equivalent to a flat connection. More generally, two connections are projectively equivalent if and only if they have the same (unparametrized) geodesics. These facts explain the relation. $\endgroup$– Robert BryantCommented Feb 19, 2013 at 18:09
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$\begingroup$ @RobertBryant, does every manifold with a projective structure (defined in terms of charts) admit a compatible (i.e., inducing the structure) projectively flat connection ? $\endgroup$– alvarezpaivaCommented Aug 26, 2017 at 8:32
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1$\begingroup$ @alvarezpaiva: Yes. In the smooth case, if you fix a volume form $\Omega$ on the manifold (or, in the non-orientable case, a nonvanishing density), there will be a unique (torsion-free) projectively flat connection for which the volume form is parallel and that is compatible with the projective structure (defined in terms of charts). There are global counterexamples in the holomorphic case. $\endgroup$– Robert BryantCommented Aug 26, 2017 at 9:00
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1$\begingroup$ @alvarezpaiva: Actually, I just realized that Vladimir's answer below already answers your question. My comment above is merely a summary of his answer, except for the brief mention of the fact that things are different in the holomorphic category. Specifically, the Riemann sphere, $\mathbb{CP}^1$ has a (unique) holomorphic projectively flat structure, but it doesn't admit a holomorphic affine connection at all. $\endgroup$– Robert BryantCommented Aug 26, 2017 at 9:53
1 Answer
I assume you are asking why two definitions of the projective structures, one given in terms of atlas, and another given as the existence of projectively flat connection, coincide.
If two connections are projectively equivalent, then their geodesics (considered as unparameterized curves) coincide, and vice versa. If a connection is projectively equivalent (locally, in a neighborhood of each point) with a flat connection, then therefore in a certain coordindate coordinate chart its geodesics are straight lines. We make an atlas from such coordinate charts. Since a local mapping that sends straight lines to straight lines is a restriction of a projective transformation, we have the existence of an atlas of charts from the open subsets of manifold to the projective space , such that transitions maps are locally the elements of the projective general linear group.
Now, if there exists such an atlas, then there exists a projectively flat connection. One of the way to see it is as follows. Take a volume form $\Omega$ on the manifold (assumed oriented but the proof can be generalised for nonorineted manifolds). Now, it is an easy exercise to see that for any connection $D$ there exists the unique projectively equivalent connection $D'(X,Y) = D(X,Y) + F(X)Y+F(Y)X$ such that the volume form is parallel w.r.t. to this connection. In the charts from your atlas consider the projectively flat connection such that $\Omega$ is parallel. Because of uniqueness, it does not depend on the coordinate chart and is therefore a globally defined projectively flat connection.
P.S. You definition of projective equivalence of two connections is nonstandard: one does not require usually that the 1-form $F$ is closed. It does not affect the proof above though.
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$\begingroup$ Thanks Vladimir . yes , you are right. 1-form doesn't need to be closed. could you please tell me where I can study more about this? $\endgroup$– DAVIDCommented Feb 20, 2013 at 14:21
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$\begingroup$ The classical sources on projectively equivalent connections are Levi-Civita: Sulle trasformazioni delle equazioni dinamiche. Ann. di Mat., serie 2a. 24, 55–300 (1896). English transl. in Regular and Chaotic Dynamics 14, 580–614 (2009) and Thomas, T. Y.: On the projective and equi-projective geo metries of paths. Proc. Natl. Acad. Sci. USA 11, 199–203 (1925) In recent time projectively equivalent connections were actively studied in the framework of parabolic geometries; see the survey M. Eastwood, Notes on projective differential geometry, arXiv:0806.3998 Everything IMHO $\endgroup$ Commented Feb 21, 2013 at 14:34