2 Projective and conformal structures
2.1 Preliminaries
Throughout the article, all manifolds are assumed to be connected, have empty boundary and unless stated otherwise, all manifolds and maps are assumed to be smooth, i.e., \(C^{\infty}\). Also, we adhere to the convention of summing over repeated indices.
2.1.1 Notation
For \(\mathbb{F}=\mathbb{R},\mathbb{C}\) the field of real or complex numbers, we denote by \(\mathbb{F}^n\) the space of column vectors of height \(n\) and by \(\mathbb{F}_n\) the space of row vectors of length \(n\) whose entries are elements of \(\mathbb{F}\). Also, we denote by \(\mathbb{FP}^2=\left(\mathbb{F}^3\setminus\{0\}\right)/\mathbb{F}^*\) the space of one-dimensional linear subspaces in \(\mathbb{F}^3\), that is, the real or complex projective plane. We denote by \(\mathbb{S}^2=\left(\mathbb{R}^3\setminus\{0\}\right)/\mathbb{R}^+\) the space of oriented one-dimensional linear subspaces in \(\mathbb{R}^3\), that is, the projective \(2\)-sphere. Likewise, we write \(\mathbb{FP}_{2}=\left(\mathbb{F}_3\setminus\{0\}\right)/\mathbb{F}^*\) for the dual (real or complex) projective plane and \(\mathbb{S}_2=\left(\mathbb{R}_3\setminus\{0\}\right)/\mathbb{R}^+\) for the dual projective \(2\)-sphere. For a non-zero vector \(x \in \mathbb{F}^3\) we write \([x]\) for its corresponding point in \(\mathbb{FP}^2\) and for a non-zero vector \(\xi \in \mathbb{F}_3\) we write \([\xi]\) for its corresponding point in \(\mathbb{FP}_2\). For non-zero vectors \(x\in\mathbb{R}^3\) and \(\xi \in \mathbb{R}_3\) we also use the notation \([x]_+\) and \([\xi]_+\) to denote the corresponding points in \(\mathbb{S}^2\) and \(\mathbb{S}_2\). Finally, we use the notation \(F(\mathbb{F}_3)\) to denote the space of complete flags in \(\mathbb{F}_3\) whose points are pairs \((\ell,\Pi)\) with \(\Pi\) being an \(\mathbb{F}\) two-dimensional linear subspace of \(\mathbb{F}_3\) containing the line \(\ell\).
2.1.2 The coframe bundle
Recall that the coframe bundle of an \(n\)-manifold \(M\) is the bundle \(\upsilon : F(T^*M) \to M\) whose fibre at a point \(p \in M\) consists of the linear isomorphisms \(u : T_pM \to \mathbb{R}^n\). The group \(\mathrm{GL}(n,\mathbb{R})\) acts transitively from the right on each \(\upsilon\)-fibre by the rule \(R_a(u)=u\cdot a =a^{-1}\circ u\) for all \(a \in \mathrm{GL}(n,\mathbb{R})\). This action turns \(\upsilon : F(T^*M) \to M\) into a principal right \(\mathrm{GL}(n,\mathbb{R})\)-bundle. The coframe bundle is equipped with a tautological \(\mathbb{R}^n\)-valued \(1\)-form \(\omega=(\omega^i)\) defined by \(\omega_u=u\circ \upsilon^{\prime}_u\). Note that \(\omega\) satisfies the equivariance property \(R_a^*\omega=a^{-1}\omega\) for all \(a \in \mathrm{GL}(n,\mathbb{R})\). The exterior derivative of local coordinates \(x : U \to \mathbb{R}^n\) on \(M\) defines a natural section \(\tilde{x} : U \to F(T^*M)\) having the reproducing property \(\tilde{x}^*\omega=\mathrm{d}x\). We will henceforth write \(F\) instead of \(F(T^*M)\) whenever \(M\) is clear from the context.
2.1.3 Associated bundles
Throughout the article we will frequently make use of the notion of an associated bundle of a principal bundle. The reader will recall that if \(\pi : P \to M\) is a principal right \(\mathrm{G}\)-bundle and \((\rho,N)\) a pair consisting of a manifold \(N\) and a homomorphism \(\rho : \mathrm{G} \to \mathrm{Diff}(N)\) into the diffeomorphism group of \(N\), then we obtain an associated fibre bundle with typical fibre \(N\) and structure group \(\mathrm{G}\) whose total space is \(P\times_{\rho} N\), that is, the elements of \(P\times_{\rho} N\) are pairs \((u,p)\) subject to the equivalence relation \[(u_1,p_1)\sim (u_2,p_2) \iff u_2=u_1\cdot g, \quad p_2=\rho(g^{-1})(p_1), \quad g \in \mathrm{G}.\] A section \(s\) of \(P\times_{\rho} N\) is then given by a map \(\sigma_{s} : P \to N\) which is equivariant with respect to the \(\mathrm{G}\)-right action on \(P\) and the right action of \(\mathrm{G}\) on \(N\) induced by \(\rho\). We say that \(s\) is represented by \(\sigma_{s}\). If \(N\) is an affine/linear space and the \(\mathrm{G}\)-action induced by \(\rho\) is affine/linear, then the associated bundle is an affine/vector bundle.
2.2 Projective structures
Recall that the set \(\mathfrak{A}(M)\) of torsion-free connections on the tangent bundle of an \(n\)-manifold \(M\) is the space of sections of an affine bundle \(\mathrm{A}(M) \to M\) of rank \(\frac{1}{2}n^2(n+1)\) which is modelled on the vector bundle \(V=S^2(T^*M)\otimes TM\). We have a canonical trace mapping \(\operatorname{tr}: V \to T^*M\) as well as an inclusion \[\iota : T^*M \to V, \quad \nu \mapsto \nu\otimes\mathrm{Id}+\mathrm{Id}\otimes \nu.\] For every \(v \in V\) we let \(v_0\) denote its trace-free part, so that \[v_0=v-\frac{1}{(n+1)}\iota(\operatorname{tr}v).\] A projective structure \(\mathfrak{p}\) on a manifold \(M\) of dimension \(n>1\) is an equivalence class of torsion-free connections on \(TM\), where two connections are declared to be equivalent if they share the same unparametrised geodesics. Weyl [44] observed the following:
Two torsion-free connections \(\nabla\) and \(\nabla^{\prime}\) on \(TM\) are projectively equivalent if and only if \((\nabla-\nabla^{\prime})_0=0\).
Consequently, the set \(\mathfrak{P}(M)\) of projective structures on \(M\) is the space of sections of an affine bundle \(\mathrm{P}(M) \to M\) of rank \(\frac{1}{2}(n+2)n(n-1)\) which is modelled on the traceless part \(V_0\) of the vector bundle \(V\). We will use the notation \(\mathfrak{p}(\nabla)\) for the projective structure \(\mathfrak{p}\) that is defined by a connection \(\nabla\). A consequence of Weyl’s result is that the set of representative connections of a projective structure \(\mathfrak{p}\) is an affine subspace \(\mathfrak{A}_{\mathfrak{p}}(M)\subset \mathfrak{A}(M)\) of the space of torsion-free connections which is modelled on the space of \(1\)-forms on \(M\).
2.3 Conformal structures
A conformal structure on a manifold \(M\) of dimension \(n>1\) is an equivalence class \([g]\) of Riemannian metrics on \(M\), where two metrics \(g\) and \(\hat{g}\) are declared to be equivalent if there exists a smooth function \(f\) on \(M\) so that \(\hat{g}=\mathrm{e}^{2f} g\). Equivalently, a conformal structure \([g]\) on \(M\) is a (smooth) choice of a coframe for every point \(p\) in \(M\), well defined up to orthogonal transformation and scaling. Consequently, the set \(\mathfrak{C}(M)\) of conformal structures on \(M\) is the space of sections of \(\mathrm{C}(M)=F/\left(\mathbb{R}^+\times\mathrm{O}(n)\right) \to M\), where \(\mathbb{R}^+\times \mathrm{O}(n)\) is the subgroup of \(\mathrm{GL}(n,\mathbb{R})\) consisting of matrices \(a\) having the property that \(aa^t\) is a non-zero multiple of the identity matrix.
A torsion-free connection \(\nabla\) on \(TM\) is called a Weyl connection or conformal connection for the conformal structure \([g]\) on \(M\) if the parallel transport maps of \(\nabla\) are angle-preserving with respect to \([g]\). A torsion-free connection \(\nabla\) is \([g]\)-conformal if for some (and hence any) representative metric \(g \in [g]\) there exists a \(1\)-form \(\beta\) on \(M\) such that \[\nabla g= 2 \beta \otimes g.\] It is a simple consequence of Koszul’s identity that the \([g]\)-conformal connections are of the form \[\tag{2.1} {}^{(g,\beta)}\nabla={}^g\nabla+g\otimes \beta^{\sharp}-\beta\otimes\mathrm{Id}-\mathrm{Id}\otimes \beta,\] where \(g\in [g]\), \(\beta\) is a \(1\)-form on \(M\) with \(g\)-dual vector field \(\beta^{\sharp}\) and \({}^g\nabla\) denotes the Levi-Civita connection of \(g\). Consequently, the set of \([g]\)-conformal connections defines an affine subspace \(\mathfrak{A}_{[g]}(M)\subset \mathfrak{A}(M)\) which is modelled on the space of \(1\)-forms on \(M\) as well. For later usage we also record that for every smooth function \(f\) on \(M\) we have \[{}^{(\exp(2f)g,\beta +\mathrm{d}f)}\nabla={}^{(g,\beta)}\nabla,\] as the reader may easily verify using the identity [3] \[\tag{2.2} {}^{\exp(2f)g}\nabla={}^g\nabla-g\otimes{}^g\nabla f+\iota(\mathrm{d}f).\] In particular, if \(\beta\) is exact, so that \(\beta=\mathrm{d}f\) for some smooth function \(f\) on \(M\), then \({}^{(g,\beta)}\nabla={}^{\exp(-2f)g}\nabla\) and hence the conformal connection determined by \((g,\beta)\) is the Levi-Civita connection of the metric \(\mathrm{e}^{-2f}g\).
We also use the notation \({}^{[g]}\nabla\) for a connection preserving the conformal structure \([g]\).
2.4 Compatibility of projective and conformal structures
Since both projective – and conformal structures give rise to affine subspaces of \(\mathfrak{A}(M)\) of the same type, we may ask how two such spaces intersect.
Let \([g]\) be a conformal – and \(\mathfrak{p}\) a projective structure on \(M\). Then \(\mathfrak{A}_{[g]}(M)\) and \(\mathfrak{A}_{\mathfrak{p}}(M)\) intersect in at most one point.
Proof. Suppose the \([g]\)-conformal connections \({}^{[g]}\nabla\) and \({}^{[g]}\hat{\nabla}\) are both elements in \(\mathfrak{A}_{\mathfrak{p}}(M)\). Then, by Lemma 2.1, there exists a \(1\)-form \(\Upsilon\) on \(M\) so that \[{}^{[g]}\nabla={}^{[g]}\hat{\nabla}+\iota(\Upsilon).\] Fixing a Riemannian metric \(g\) defining \([g]\), we also have \(1\)-forms \(\beta,\hat{\beta}\) on \(M\) so that \[{}^{[g]}\nabla={}^g\nabla+g\otimes \beta^{\sharp}-\iota(\beta)\quad \text{and}\quad{}^{[g]}\hat{\nabla}={}^g\nabla+g\otimes \hat{\beta}^{\sharp}-\iota(\hat{\beta}).\] Applying these formulae we obtain \[\iota(\Upsilon+\beta-\hat{\beta})=g\otimes\left(\beta^{\sharp}-\hat{\beta}^{\sharp}\right).\] Taking the trace gives \[(n+1)\left(\Upsilon+\beta-\hat{\beta}\right)=\beta-\hat{\beta},\] so that \(\Upsilon=-\frac{n}{(n+1)}(\beta-\hat{\beta})\). Therefore we must have \[\iota\left(\beta-\hat{\beta}\right)=(n+1)g\otimes \left(\beta^{\sharp}-\hat{\beta}^{\sharp}\right).\] Contracting this last equation with the dual metric \(g^\sharp\) implies \[0=(n+2)(n-1)\left(\beta^{\sharp}-\hat{\beta}^{\sharp}\right),\] so that \(\beta=\hat{\beta}\) provided \(n>1\). It follows that \(\Upsilon\) vanishes too, therefore \({}^{[g]}\nabla={}^{[g]}\hat{\nabla}\), as claimed.
Lemma 2.2 raises the question whether or not one can still determine a unique point \({}^{[g]}\nabla\in \mathfrak{A}_{[g]}(M)\) and a unique point \(\nabla \in \mathfrak{A}_{\mathfrak{p}}(M)\) in the general case, where \(\mathfrak{A}_{[g]}(M)\) and \(\mathfrak{A}_{\mathfrak{p}}(M)\) might not intersect. Formally speaking, we are interested in maps \[\psi=\left(\psi^1,\psi^2\right) : \mathfrak{P}(M) \times \mathfrak{C}(M) \to \mathfrak{A}(M)\times \mathfrak{A}(M)\] satisfying the following properties:
\(\psi^1(\mathfrak{p},[g]) \in \mathfrak{A}_{\mathfrak{p}}(M)\) and \(\psi^2(\mathfrak{p},[g]) \in \mathfrak{A}_{[g]}(M)\);
If \(\mathfrak{A}_{\mathfrak{p}}(M)\cap \mathfrak{A}_{[g]}(M)\) is non-empty, then \(\psi^2(\mathfrak{p},[g])-\psi^1(\mathfrak{p},[g])=0\);
\(\psi\) is equivariant with respect to the natural right action of the diffeomorphism group \(\mathrm{Diff}(M)\) on \(\mathfrak{P}(M)\times \mathfrak{C}(M)\) and \(\mathfrak{A}(M)\times \mathfrak{A}(M)\).
We will next discuss a geometrically natural and explicit map \(\psi\) having these properties.
To this end let \(g\) be a Riemannian metric on \(M\) and \(\nabla\) a torsion-free connection on \(TM\). Consider the first-order differential operator for \(g\) mapping into the space of \(1\)-forms on \(M\) with values in \(\mathrm{End}(TM)\) \[\tag{2.3} g \mapsto A_{[g]}=\left(\nabla-{}^g\nabla-g\otimes X_g\right)_0,\] where \(X_g \in \Gamma(TM)\) is \[\tag{2.4} X_g=\frac{(n+1)}{(n+2)(n-1)}\,\operatorname{tr}\left(g^{\sharp}\otimes (\nabla-{}^g\nabla)_0\right).\] The following result is essentially contained in [35] – except for (vi). For the convenience of the reader we include a proof.
The \(1\)-form \(A_{[g]}\) has the following properties:
the endomorphism \(A_{[g]}(X)\) is trace-free for all \(X \in \Gamma(TM)\);
for all \(X,Y \in \Gamma(TM)\) we have \(A_{[g]}(X)Y=A_{[g]}(Y)X\);
\(A_{[g]}\) only depends on the projective equivalence class of \(\nabla\);
\(A_{[g]}\) only depends on the conformal equivalence class of \(g\);
\(A_{[g]}\equiv 0\) if and only if there exists a \([g]\)-conformal connection which is projectively equivalent to \(\nabla\);
for \(n=2\) the endomorphism \(A_{[g]}(X)\) is symmetric with respect to \([g]\) for all \(X \in \Gamma(TM)\);
Proof. The properties (i) and (ii) are obvious from the definition.
(iii) Recall that two affine torsion-free connections \(\nabla\) and \(\hat{\nabla}\) are projectively equivalent if and only if \((\nabla-\hat{\nabla})_0=0\). The claim follows from the linearity of the “taking the trace-free part” operation.
(iv) Let \(\hat{g}=\mathrm{e}^{2f}g\) for some smooth real-valued function \(f\) on \(M\). Then we have \[{}^{\hat{g}}\nabla={}^g\nabla-g\otimes{}^g\nabla f+\iota(\mathrm{d}f)\] and hence \[\begin{aligned} \left(\nabla-{}^{\hat{g}}\nabla\right)_0&=\left(\nabla-{}^g\nabla\right)_0+\left(g\otimes {}^g\nabla f-\iota(\mathrm{d}f)\right)_0\\ &=\left(\nabla-{}^g\nabla\right)_0+\left(g\otimes {}^g\nabla f\right)_0\\ &=\left(\nabla-{}^g\nabla\right)_0+g\otimes {}^g\nabla f-\frac{1}{(n+1)}\iota(\mathrm{d}f). \end{aligned}\] We obtain \[\begin{aligned} X_{\hat{g}}&=\frac{(n+1)}{(n+2)(n-1)}\operatorname{tr}\left[\hat{g}^{\sharp}\otimes \left(\left(\nabla-{}^g\nabla\right)_0+g\otimes {}^g\nabla f-\frac{1}{(n+1)}\iota(\mathrm{d}f)\right)\right]\\ &=\mathrm{e}^{-2f}\left(X_g+\frac{n(n+1)}{(n+2)(n-1)}{}^g\nabla f-\frac{2}{(n+2)(n-1)}{}^g\nabla f\right)\\ &=\mathrm{e}^{-2f}\left(X_g+{}^g\nabla f\right). \end{aligned}\] This gives \[\begin{aligned} {}^{\hat{g}}\nabla+\hat{g}\otimes X_{\hat{g}}&={}^g\nabla-g\otimes{}^g\nabla f+\iota(\mathrm{d}f)+\mathrm{e}^{2f}g\otimes\mathrm{e}^{-2f}\left(X_g+{}^g\nabla f\right)\\ &={}^g\nabla+g\otimes X_g+\iota(\mathrm{d}f), \end{aligned}\] so that \[\left({}^{\hat{g}}\nabla+\hat{g}\otimes X_{\hat{g}}\right)_0=\left({}^g\nabla+g\otimes X_g\right)_0,\] which shows that \(A_{[g]}\) does indeed only depend on the conformal class of \(g\).
(v) Recall that the \({[g]}\)-conformal connections are of the form \[{}^{[g]}\nabla={}^g\nabla+g\otimes \beta^{\sharp}-\iota(\beta),\] where \(g\) is any metric in the conformal class \({[g]}\) and \(\beta\) is some \(1\)-form on \(M\). Therefore we have \[\left({}^{[g]}\nabla-{}^{g}\nabla\right)_0=\left(g\otimes\beta^{\sharp}\right)_0=g\otimes \beta^{\sharp}-\frac{1}{(n+1)}\iota (\beta)\] and thus as before we compute that \(X_g=\beta^{\sharp}\). We obtain \[\begin{aligned} A_{[g]}&=\left[{}^{[g]}\nabla-\left({}^g\nabla+g\otimes X_g\right)\right]_0\\ &=\left[{}^g\nabla+g\otimes \beta^{\sharp}-\iota(\beta)-{}^g\nabla-g\otimes \beta^{\sharp}\right]_0=\left[-\iota(\beta)\right]_0=0. \end{aligned}\] Conversely, suppose \(\mathfrak{p}\) is a projective structure for which there exists a conformal structure \({[g]}\) with \(A_{[g]}\equiv 0\). Fixing a Riemannian metric \(g \in [g]\) and a \(\mathfrak{p}\)-representative connection \(\nabla\), we must have \[\nabla-({}^g\nabla+g \otimes X_g)=\iota(\beta),\] for some \(1\)-form \(\beta\) on \(M\). Adding \(\iota((X_g)^{\flat})\) gives \[\nabla-\left({}^g\nabla+g\otimes X_g-\iota\left((X_g)^{\flat}\right)\right)=\iota\left(\beta+(X_g)^{\flat}\right),\] so that Lemma 2.1 implies that \(\nabla\) and the \([g]\)-conformal connection \[{}^g\nabla+g\otimes X_g-\iota\left((X_g)^{\flat}\right)\] are projectively equivalent.
(vi) Let now \(n=2\). We need to show that for \(g \in [g]\) and all vector fields \(X,Y,Z \in \Gamma(TM)\), we have \[g(A_{[g]}(X)Y,Z)=g(Y,A_{[g]}(X)Z).\] Without losing generality, we can assume that locally \(g=(\mathrm{d}x^1)^2+(\mathrm{d}x^2)^2\) for coordinates \(x=(x^1,x^2) : U \to \mathbb{R}^2\) on \(M\). Let \(\Gamma^i_{jk}\) denote the Christoffel symbols of \(\nabla\) with respect to \(x\). Since the Christoffel symbols of \({}^g\nabla\) vanish identically on \(U\), we obtain with a simple calculation \[X_g=-\frac{3}{4}\left(w_1+w_3\right)\frac{\partial}{\partial x^1}+\frac{3}{4}\left(w_0+w_2\right)\frac{\partial}{\partial x^2},\] where \[w_0=\Gamma^2_{11}, \quad 3w_1=-\Gamma^1_{11}+2\Gamma^2_{12}, \quad 3w_2=-2\Gamma^1_{12}+\Gamma^2_{22}, \quad w_3=-\Gamma^1_{22}.\] Likewise, we compute \[\begin{gathered} A_{[g]}=\frac{1}{2}\left(a_1e^{11}_{\phantom{11}1}-a_2e^{11}_{\phantom{11}2}-a_2e^{12}_{\phantom{12}1}-a_1e^{12}_{\phantom{12}2}\right.\\-\left.a_2e^{21}_{\phantom{21}1}-a_1e^{21}_{\phantom{21}2}-a_1e^{22}_{\phantom{22}1}+a_2e^{22}_{\phantom{22}2}\right)\end{gathered}\] where we write \(e^{ij}_{\phantom{ij}k}=\mathrm{d}x^i\otimes \mathrm{d}x^j\otimes \frac{\partial}{\partial x^k}\) and \[a_1=\frac{1}{2}(w_3-3w_1), \quad a_2=\frac{1}{2}(3w_2-w_0).\] The claim follows from an elementary calculation.
By construction, the \(1\)-form \(A_{[g]}\) vanishes identically if and only if \(\nabla\) is projectively equivalent to a conformal connection. The necessary and sufficient conditions for a torsion-free connection to be projectively equivalent to a Levi-Civita connection were given in [6]. The reader may also consult [7] for the role of Einstein metrics in projective differential geometry.
As a corollary to Theorem 2.4 and Lemma 2.2 we obtain the following result.
For every conformal structure \([g]\) on the projective manifold \((M,\mathfrak{p})\), there exists a unique \([g]\)-conformal connection \({}^{[g]}\nabla\) so that \({}^{[g]}\nabla+A_{[g]}\in \mathfrak{p}\).
Note that Corollary 2.6 provides a unique point \({}^{[g]}\nabla\in \mathfrak{A}_{[g]}(M)\) and a unique point \({}^{[g]}\nabla+A_{[g]} \in \mathfrak{A}_{\mathfrak{p}}(M)\). We may define \[\psi\left(\mathfrak{p},[g]\right)=\left({}^{[g]}\nabla+A_{[g]},{}^{[g]}\nabla\right).\] Since the map which sends a Riemannian metric to its Levi-Civita connection is equivariant with respect to the action of \(\mathrm{Diff}(M)\) on the space of Riemannian metrics and on \(\mathfrak{A}(M)\), it follows that the map \(\psi\) has all the properties listed in Remark 2.3.
Proof of Corollary 2.6. Let \(\nabla\) be a connection defining \(\mathfrak{p}\) and \(g\) a smooth metric defining \([g]\). Set \[{}^{[g]}\nabla={}^g\nabla+g\otimes X_g-(X_g)^{\flat}\otimes \mathrm{Id}-\mathrm{Id}\otimes (X_g)^{\flat},\] where \(X_g\) is defined as before (see (2.4)). Then, property (i) of \(A_{[g]}\) proved in Theorem 2.4 implies that \[\left(\nabla-\left({}^{[g]}\nabla+A_{[g]}\right)\right)_0=\left(\nabla-({}^g\nabla+g\otimes X_g)\right)_0-A_{[g]}=A_{[g]}-A_{[g]}=0,\] so that \({}^{[g]}\nabla+A_{[g]}\) is projectively equivalent to \(\nabla\) by Lemma 2.1. If \({}^{[g]}\nabla^{\prime}\) is another \([g]\)-conformal connection so that \({}^{[g]}\nabla^{\prime}+A_{[g]}\) defines \(\mathfrak{p}\), then \[\left({}^{[g]}\nabla-{}^{[g]}\nabla^{\prime}\right)_0=0,\] hence \({}^{[g]}\nabla={}^{[g]}\nabla^{\prime}\) by Lemma 2.2.
2.5 A diffeomorphism invariant functional
We will henceforth assume \(M\) to be oriented. For a pair \((\mathfrak{p},[g])\) consisting of a projective structure and a conformal structure on \(M\), we consider the non-negative \(n\)-form \(|A_{[g]}|^n_gd\mu_g\), where \(g\) is any metric defining \([g]\), the \(n\)-form \(d\mu_g\) denotes its volume form and where \(A_{[g]}\) is computed with respect to \(\mathfrak{p}\). For \(f \in C^{\infty}(M)\) we have \[|A_{[g]}|_{\mathrm{e}^{2f}g}=\mathrm{e}^{-f}|A_{[g]}|_{g} \quad \text{and} \quad d\mu_{\mathrm{e}^{2f}g}=e^{nf}d\mu_g,\] it follows that \(|A_{[g]}|^n_gd\mu_g\) depends only on the conformal structure \([g]\). Consequently, we obtain a non-negative functional \[\mathcal{F} : \mathfrak{P}(M)\times \mathfrak{C}(M) \to \mathbb{R}^{+}_{0}\cup \{\infty\}, \quad (\mathfrak{p},[g]) \mapsto \int_{M} |A_{[g]}|^n_gd\mu_g.\] By construction, \(\mathcal{F}\) is invariant under simultaneous action of \(\mathrm{Diff}(M)\) on \(\mathfrak{P}(M)\) and \(\mathfrak{C}(M)\).
We may also fix a projective structure \(\mathfrak{p}\) on \(M\) and define \(\mathcal{E}_{\mathfrak{p}}=\mathcal{F}[(\mathfrak{p},\cdot)]\) which is a functional on \(\mathfrak{C}(M)\) only. We may study the infimum of \(\mathcal{E}_{\mathfrak{p}}\) among all conformal structures on \(M\), and ask whether there is actually a minimising conformal structure which achieves this infimum. The infimum \[\Gamma\delta(M,\mathfrak{p}):=\inf_{[g] \in \mathfrak{C}(M)}\mathcal{E}_{\mathfrak{p}}([g]),\] which may be considered as a measure of how far \(\mathfrak{p}\) deviates from being defined by a conformal connection, is a new global invariant for oriented projective manifolds. Note that reversing the role of \(\mathfrak{p}\) and \([g]\) does not give us a global invariant for conformal manifolds. Clearly, fixing a conformal structure and considering the infimum over \(\mathfrak{P}(M)\) yields zero for every choice of conformal structure \([g]\).
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