I) Here we discuss the problem of defining a connection on a conformal manifold $M$. We start with a conformal class $[g_{\mu\nu}]$ of globally$^{1}$ defined metrics 

$$\tag{1} g^{\prime}_{\mu\nu}~=~\Omega^2 g_{\mu\nu}$$

given by Weyl transformations/rescalings. Under mild assumption about the manifold $M$ (para-compactness), we may assume that there exists a conformal class $[A_{\mu}]$ of globally defined co-vectors/one-forms connected via Weyl transformations as

$$\tag{2} A^{\prime}_{\mu}~=~A_{\mu} + \partial_{\mu}\ln(\Omega^2). $$

In particular it is implicitly understood that a Weyl transformation [of a pair $(g_{\mu\nu},A_{\mu})$ of representatives] act in tandem/is synchronized with the same globally defined function $\Omega$ in eqs. (1) and (2) simultaneously.

II) Besides Weyl transformations, we can act (in the active picture) with diffeomorphisms. Locally, in the passive picture, the pair $(g_{\mu\nu},A_{\mu})$ transforms as covariant tensors

$$ \tag{3} g_{\mu\nu}~=~ \frac{\partial  x^{\prime \rho}}{\partial x^{\mu}} g^{\prime}_{\rho\sigma}\frac{\partial  x^{\prime \sigma}}{\partial x^{\nu}},  $$

$$ \tag{4} A_{\mu}~=~ \frac{\partial  x^{\prime \nu}}{\partial x^{\mu}} A^{\prime}_{\nu}. $$

under general coordinate transformations 

$$ \tag{5} x^{\mu} ~\longrightarrow~ x^{\prime \nu}~= ~f^{\nu}(x). $$

III) We next introduce the unique torsionfree tangent-space Weyl connection $\nabla$ with corresponding Christoffel symbols $\Gamma^{\lambda}_{\mu\nu}$ that covariantly preserves the metric in the following sense:

$$ \tag{6} (\nabla_{\lambda}-A_{\lambda})g_{\mu\nu}~=~0. $$

The Weyl connection $\nabla$ and its Christoffel symbols $\Gamma^{\lambda}_{\mu\nu}$ are independent of the pair $(g_{\mu\nu},A_{\mu})$ of representatives within the conformal class $[(g_{\mu\nu},A_{\mu})]$. (But the construction depends of course on the conformal class $[(g_{\mu\nu},A_{\mu})]$.) In other words, the Weyl Christoffel symbols are invariant under Weyl transformations 

$$ \tag{7} \Gamma^{\prime\lambda}_{\mu\nu}~=~\Gamma^{\lambda}_{\mu\nu}.$$ 

The lowered Weyl Christoffel symbols are uniquely given by

$$ \Gamma_{\lambda,\mu\nu}~=~g_{\lambda\rho}  \Gamma^{\rho}_{\mu\nu} $$
$$ ~=~\frac{1}{2}\left((\partial_{\mu}-A_{\mu})g_{\nu\lambda} +(\partial_{\nu}-A_{\nu})g_{\mu\lambda}-(\partial_{\lambda}-A_{\lambda})g_{\mu\nu} \right) $$
$$\tag{8} ~=~\Gamma^{(g)}_{\lambda,\mu\nu}+\frac{1}{2}\left(A_{\mu}g_{\nu\lambda}-A_{\nu}g_{\mu\lambda}+A_{\lambda}g_{\mu\nu} \right), $$

where $\Gamma^{(g)}_{\lambda,\mu\nu}$ denote the lowered Levi-Civita Christoffel symbols for the representative $g_{\mu\nu}$. 
The lowered Weyl Christoffel symbols $\Gamma_{\lambda,\mu\nu}$ scale under Weyl transformations as

$$ \tag{9} \Gamma^{\prime}_{\lambda,\mu\nu}~=~\Omega^2\Gamma_{\lambda,\mu\nu}.$$ 

The corresponding determinant bundle has a Weyl connection given by

$$ \tag{10} \Gamma_{\lambda}~=~\Gamma^{\mu}_{\lambda\mu}~=~(\partial_{\lambda}-A_{\lambda})\ln \sqrt{\det(g_{\mu\nu})}.$$

IV) Let us next define a conformal class $[\rho]$ of a density $\rho$ of weights $(w,h)$, who scales under Weyl transformations as

$$  \tag{11} \rho^{\prime}~=~ \Omega^w\rho $$

with Weyl weight $w$, and as a density

$$\tag{12} \rho^{\prime}~=~\frac{\rho}{J^h}$$

of weight $h$ under general coordinate transformations (5). Here

$$\tag{13} J ~:=~\det(\frac{\partial x^{\prime \nu}}{\partial x^{\mu}}) $$

is the Jacobian. 

Example: The determinant $\det(g_{\mu\nu})$ is a density with $h=2$ and $w=2d$, where $d$ is the dimension of the manifold $M$.

V) The concept of (conformal classes of) densities $\rho$ of weights $(w,h)$ can be generalized to (conformal classes of) tensor densities $T^{\mu_1\ldots\mu_m}_{\nu_1\ldots\nu_n}$ of weights $(w,h)$ in a straightforward manner. For instance, a vector density  of weights $(w,h)$ transforms as

$$ \tag{14} \xi^{\prime \mu}~=~ \frac{1}{J^h}\frac{\partial  x^{\prime \mu}}{\partial x^{\nu}} \xi^{\nu} $$

under general coordinate transformations (5), and scales as 

$$ \tag{15} \xi^{\prime \mu}~=~\Omega^w \xi^{\mu} $$

under Weyl transformations. Similarly, a co-vector density of weights $(w,h)$ transforms as

$$ \tag{16} \eta^{\prime}_{\mu}~=~ \frac{1}{J^h}\frac{\partial x^{\nu}}{\partial  x^{\prime \mu}} \eta_{\nu} $$

under general coordinate transformations (5), and scales as 

$$ \tag{17} \eta^{\prime}_{\mu}~=~\Omega^w \eta_{\mu} $$

under Weyl transformations. And so forth for arbitrary tensor densities $T^{\mu_1\ldots\mu_m}_{\nu_1\ldots\nu_n}$.

Example: The metric $g_{\mu\nu}$ is a tensor density with $h=0$ and $w=2$. The one-form $A_{\mu}$ is not a tensor density, cf. eq. (2). 

VI) Finally, one can discuss the definition of covariantly conserved (conformal classes of) tensor densities $T^{\mu_1\ldots\mu_m}_{\nu_1\ldots\nu_n}$. A density $\rho$ of weights $(w,h)$ is covariantly conserved if

$$\tag{18} (\nabla_{\lambda}-\frac{w}{2}A_{\lambda})\rho~\equiv~ (\partial_{\lambda}-h \Gamma_{\lambda}-\frac{w}{2}A_{\lambda})\rho~=~0. $$

A vector density of weights $(w,h)$ is covariantly conserved if

$$\tag{19}  (\nabla_{\lambda}-\frac{w}{2}A_{\lambda})\xi^{\mu}~\equiv~ (\partial_{\lambda}-h \Gamma_{\lambda}-\frac{w}{2}A_{\lambda})\xi^{\mu}+\Gamma_{\lambda\nu}^{\mu}\xi^{\nu} ~=~0. $$

A co-vector density of weights $(w,h)$ is covariantly conserved if

$$\tag{20}(\nabla_{\lambda}-\frac{w}{2}A_{\lambda})\eta_{\mu}~\equiv~  (\partial_{\lambda}-h \Gamma_{\lambda}-\frac{w}{2}A_{\lambda})\eta_{\mu}-\Gamma_{\lambda\mu}^{\nu}\eta_{\nu} ~=~0. $$

In particular, if $T^{\mu_1\ldots\mu_m}_{\nu_1\ldots\nu_n}$ is a tensor density of weights $(w,h)$, then the covariant derivative $(\nabla_{\lambda}-\frac{w}{2}A_{\lambda})T^{\mu_1\ldots\mu_m}_{\nu_1\ldots\nu_n}$ is also a tensor density of weights $(w,h)$.

--

$^{1}$ We ignore for simplicity the concept of locally defined conformal classes.

Is there a "covariant derivative" for conformal transformation?
I checked my internet setting today and it seem that everything has being going through some proxy server for the last few weeks.  How screwed am I?

Could they get my internet login password?  All my files on Dropbox?  I buy things online too, is my credit card compromised?

EDIT:

The IP of the proxy is 85.214.85.193, but on my other computer which was also proxied the IP was something different (I forgot to write it down).
