Projective connection

Indifferential geometry,aprojective connectionis a type ofCartan connectionon adifferentiable manifold.

The structure of a projective connection is modeled on the geometry ofprojective space,rather than theaffine spacecorresponding to anaffine connection.Much like affine connections, projective connections also definegeodesics.However, these geodesics are notaffinely parametrized.Rather they are projectively parametrized, meaning that their preferred class of parameterizations is acted upon by the group offractional linear transformations.

Like an affine connection, projective connections have associated torsion and curvature.

Projective space as the model geometry

The first step in defining any Cartan connection is to consider the flat case: in which the connection corresponds to theMaurer-Cartan formon ahomogeneous space.

In the projective setting, the underlying manifold$M$of the homogeneous space is the projective spaceRPⁿwhich we shall represent byhomogeneous coordinates$[x_{0},\dots,x_{n}]$.The symmetry group of$M$isG= PSL(n+1,R).LetHbe theisotropy groupof the point$[1,0,0,\ldots,0]$.Thus,M=G/Hpresents$M$as a homogeneous space.

Let${\mathfrak {g}}$be theLie algebraofG,and${\mathfrak {h}}$that ofH.Note that${\mathfrak {g}}={\mathfrak {s}}{\mathfrak {l}}(n+1,{\mathbb {R} })$.As matrices relative to the homogeneousbasis,${\mathfrak {g}}$consists oftrace-free$(n+1)\times (n+1)$matrices:

$\left({\begin{matrix}\lambda &v^{i}\\w_{j}&a_{j}^{i}\end{matrix}}\right),\quad (v^{i})\in {\mathbb {R} }^{1\times n},(w_{j})\in {\mathbb {R} }^{n\times 1},(a_{j}^{i})\in {\mathbb {R} }^{n\times n},\lambda =-\sum _{i}a_{i}^{i}$.

And${\mathfrak {h}}$consists of all these matrices with$(w_{j})=0$.Relative to the matrix representation above, the Maurer-Cartan form ofGis a system of1-forms$(\xi,\ Alpha _{j},\ Alpha _{j}^{i},\ Alpha ^{i})$satisfying the structural equations (written using theEinstein summation convention):

$d\xi +\ Alpha ^{i}\wedge \ Alpha _{i}=0$

$da_{j}+a_{j}\wedge \zeta +a_{j}^{k}\wedge a_{k}=0$

$da_{j}^{i}+a^{i}\wedge a_{j}+a_{k}^{i}\wedge a_{j}^{k}=0$

$da^{i}+\zeta \wedge a^{i}+a^{k}\wedge a_{k}^{i}=0$

Projective structures on manifolds

A projective structure is alinear geometryon a manifold in which two nearby points are connected by a line (i.e., an unparametrizedgeodesic) in a unique manner. Furthermore, an infinitesimal neighborhood of each point is equipped with a class ofprojective frames.According to Cartan (1924),

Une variété (ou espace) à connexion projective est une variété numérique qui, au voisinage immédiat de chaque point, présente tous les caractères d'un espace projectif et douée de plus d'une loi permettant de raccorder en un seul espace projectif les deux petits morceaux qui entourent deux points infiniment voisins....

Analytiquement, on choisira, d'une manière d'ailleurs arbitraire, dans l'espace projectif attaché à chaque pointade la variété, unrepéredéfinissant un système de coordonnées projectives.... Le raccord entre les espaces projectifs attachés à deux points infiniment voisinsaeta'se traduira analytiquement par une transformation homographique....

This is analogous to Cartan's notion of anaffine connection,in which nearby points are thus connected and have an affineframe of referencewhich is transported from one to the other (Cartan, 1923):

La variété sera dite à "connexion affine" lorsqu'on aura défini, d'une manière d'ailleurs arbitraire, une loi permettant de repérer l'un par rapport à l'autre les espaces affines attachés à deux pointsinfiniment voisinsquelconquesmetm'de la variété; cete loi permettra de dire que tel point de l'espace affine attaché au pointm'correspond à tel point de l'espace affine attaché au pointm,que tel vecteur du premier espace es parallèle ou équipollent à tel vecteur du second espace.

In modern language, a projective structure on ann-manifoldMis aCartan geometrymodelled on projective space, where the latter is viewed as a homogeneous space for PSL(n+1,R). In other words it is a PSL(n+1,R)-bundle equipped with

• a PSL(n+1,R)-connection (theCartan connection)
• areduction of structure groupto the stabilizer of a point in projective space

such that thesolder forminduced by these data is an isomorphism.

References

• Cartan, Élie (1923)."Sur les variétés à connexion affine, et la théorie de la relativité généralisée (première partie)".Annales Scientifiques de l'École Normale Supérieure.40:325–412.doi:10.24033/asens.751.
• Cartan, Élie (1924)."Sur les varietes a connexion projective".Bulletin de la Société Mathématique.52:205–241.doi:10.24033/bsmf.1053.
• Hermann, R., Appendix 1-3 in Cartan, E.Geometry of Riemannian Spaces,Math Sci Press, Massachusetts, 1983.
• Cartan, Élie (1926), "Les groupes d'holonomie des espaces généralisés",Acta Mathematica,48(1–2): 1–42,doi:10.1007/BF02629755
• Sharpe, R.W. (1997).Differential Geometry: Cartan's Generalization of Klein's Erlangen Program.Springer-Verlag, New York.ISBN0-387-94732-9.

External links

• Ü. Lumiste(2001) [1994],"Projective connection",Encyclopedia of Mathematics,EMS Press