metric of second degree
$g_{ab} = \hat{g}_{ab} exp(i (a + b_u x^u + c_{uv} x^u x^v))$
so $c_{uv}$ might as well be symmetric.
$g_{ab,c} = g_{ab} i (a + b_u x^u + c_{uv} x^u x^v)_{,c}$
$g_{ab,c} = i (b_c + 2 c_{cu} x^u) g_{ab}$

$\Gamma_{abc} = \frac{1}{2} ( g_{ab,c} + g_{ac,b} - g_{bc,a} )$
$= i \frac{1}{2} ( (b_c + 2 c_{cu} x^u) g_{ab} + (b_b + 2 c_{bu} x^u) g_{ac} - (b_a + 2 c_{au} x^u) g_{bc} )$
$= i \frac{1}{2} ( g_{ab} b_c + g_{ac} b_b - g_{bc} b_a + 2 x^u (g_{ab} c_{cu} + g_{ac} c_{bu} - g_{bc} c_{au}) )$

metric of infinite degree
$g^{(n)}_{ab} = \hat{g}_{ab} exp(i \Sigma^n_{i=0} \tilde{g}^{(i)}_{u_1 ... u_i} x^{u_1} \cdot ... \cdot x^{u_i} ))$
$\tilde{g}^{(i)}_{u_1 ... u_i}$ is symmetric on all indexes.
Let $\theta^{(n)} = \Sigma^n_{i=0} \tilde{g}^{(i)}_{u_1 ... u_i} x^{u_1} \cdot ... \cdot x^{u_i}$
$g^{(n)}_{ab} = \hat{g}_{ab} exp(i \theta^{(n)})$

$g^{(n)ab} = \hat{g}^{ab} exp(-i \theta^{(n)})$
for $\hat{g}^{ac} \hat{g}^{cb} = \delta^a_b$
so $g^{(n)ac} g^{(n)}_{cb} = \hat{g}^{ac} exp(-i \theta^{(n)}) \hat{g}_{cb} exp(i \theta^{(n)}) = \delta^a_b$

first derivative of $\theta$
$\theta^{(1)}_{,c} = (\tilde{g}^{(0)} + \tilde{g}^{(1)}_u x^u)_{,c}$
$= \tilde{g}^{(1)}_c$

$\theta^{(2)}_{,c} = (\tilde{g}^{(0)} + \tilde{g}^{(1)}_{u_1} x^{u_1} + \tilde{g}^{(2)}_{u_1 u_2} x^{u_1} x^{u_2})_{,c}$
$= \tilde{g}^{(1)}_c + \tilde{g}^{(2)}_{c u_2} x^{u_2} + \tilde{g}^{(2)}_{u_1 c} x^{u_1}$
... so long as $\tilde{g}^{(i)}$ is symmetric on all indexes ...
$= \tilde{g}^{(1)}_c + 2 \tilde{g}^{(2)}_{c u_1} x^{u_1}$

$\theta^{(n)}_{,c} = (\Sigma^n_{i=0} \tilde{g}^{(i)}_{u_1 ... u_i} x^{u_1} \cdot ... \cdot x^{u_i} )_{,c}$
$= \Sigma_{k=1}^n k \tilde{g}^{(k)}_{c \underbrace{u_1 ... u_{k-1}}} x^{u_1} \cdot ... \cdot x^{u_{k-1}}$

second derivative of $\theta$
$\theta^{(1)}_{,cd} = (\tilde{g}^{(1)}_c)_{,d} = 0$

$\theta^{(2)}_{,cd} = (\tilde{g}^{(1)}_c + 2 \tilde{g}^{(2)}_{c u_1} x^{u_1})_{,d}$
$= 2 \tilde{g}^{(2)}_{c d}$

$\theta^{(3)}_{,cd} = (\tilde{g}^{(1)}_c + 2 \tilde{g}^{(2)}_{c u_1} x^{u_1} + 3 \tilde{g}^{(3)}_{c u_1 u_2} x^{u_1} x^{u_2})_{,d}$
$= 2 \tilde{g}^{(2)}_{c d} + 6 \tilde{g}^{(3)}_{c d u_1} x^{u_1}$

$\theta^{(n)}_{,cd} = (\Sigma_{k=1}^n k \tilde{g}^{(k)}_{c \underbrace{u_1 ... u_{k-1}}} x^{u_1} \cdot ... \cdot x^{u_{k-1}})_{,d}$
$= \Sigma_{k=1}^{n-1} k (k+1) \tilde{g}^{(k+1)}_{c d \underbrace{u_1 ... u_{k-1}}} x^{u_1} \cdot ... \cdot x^{u_{k-1}}$

n-th derivative of $\theta$
$\theta^{(n)}_{,\underbrace{a_1 ... a_m}} = \Sigma_{k=1}^{n+1-m} (\Pi_{l=k}^{k+m-1} l) \tilde{g}^{(k+m-1)}_{\underbrace{a_1 ... a_m} \underbrace{u_1 ... u_{k-1}}} x^{u_1} \cdot ... x^{u_{k-1}}$

first derivative of $\theta$, squared
$\theta^{(1)}_{,c} \theta^{(1)}_{,d} = \tilde{g}^{(1)}_c \tilde{g}^{(1)}_d$

$\theta^{(2)}_{,c} \theta^{(2)}_{,d} = (\tilde{g}^{(1)}_c + \tilde{g}^{(2)}_{c u_1} x^{u_1}) (\tilde{g}^{(1)}_d + \tilde{g}^{(2)}_{d v_1} x^{v_1})$
$= \tilde{g}^{(1)}_c \tilde{g}^{(1)}_d + (\tilde{g}^{(2)}_{c u_1} \tilde{g}^{(1)}_d + \tilde{g}^{(1)}_{c} \tilde{g}^{(2)}_{d u_1}) x^{u_1} + \tilde{g}^{(2)}_{c u_1} \tilde{g}^{(2)}_{d v_1} x^{u_1} x^{v_1}$
$= \theta^{(1)}_{,c} \theta^{(1)}_{,d} + (\tilde{g}^{(2)}_{c u_1} \tilde{g}^{(1)}_d + \tilde{g}^{(1)}_{c} \tilde{g}^{(2)}_{d u_1}) x^{u_1} + \tilde{g}^{(2)}_{c u_1} \tilde{g}^{(2)}_{d v_1} x^{u_1} x^{v_1}$

$\theta^{(3)}_{,c} \theta^{(3)}_{,d} = (\tilde{g}^{(1)}_c + \tilde{g}^{(2)}_{c u_1} x^{u_1} + \tilde{g}^{(3)}_{c u_1 u_2} x^{u_1} x^{u_2}) (\tilde{g}^{(1)}_d + \tilde{g}^{(2)}_{d v_1} x^{v_1} + \tilde{g}^{(3)}_{d v_1 v_2} x^{v_1} x^{v_2})$
$= \tilde{g}^{(1)}_c \tilde{g}^{(1)}_d + (\tilde{g}^{(2)}_{c u_1} \tilde{g}^{(1)}_d + \tilde{g}^{(1)}_{c} \tilde{g}^{(2)}_{d u_1}) x^{u_1} + (\tilde{g}^{(2)}_{c u_1} \tilde{g}^{(2)}_{d u_2} + \tilde{g}^{(3)}_{c u_1 u_2} \tilde{g}^{(1)}_d + \tilde{g}^{(1)}_c \tilde{g}^{(3)}_{d u_1 u_2}) x^{u_1} x^{u_2} + (\tilde{g}^{(2)}_{c u_1} \tilde{g}^{(3)}_{d u_2 u_3} + \tilde{g}^{(3)}_{c u_1 u_2} \tilde{g}^{(2)}_{d u_3}) x^{u_1} x^{u_2} x^{u_3} + \tilde{g}^{(3)}_{c u_1 u_2} \tilde{g}^{(3)}_{d u_3 u_4} x^{u_1} x^{u_2} x^{u_3} x^{u_4} $
$= \theta^{(2)}_c \theta^{(2)}_d + (\tilde{g}^{(3)}_{c u_1 u_2} \tilde{g}^{(1)}_d + \tilde{g}^{(1)}_c \tilde{g}^{(3)}_{d u_1 u_2}) x^{u_1} x^{u_2} + (\tilde{g}^{(2)}_{c u_1} \tilde{g}^{(3)}_{d u_2 u_3} + \tilde{g}^{(3)}_{c u_1 u_2} \tilde{g}^{(2)}_{d u_3}) x^{u_1} x^{u_2} x^{u_3} + \tilde{g}^{(3)}_{c u_1 u_2} \tilde{g}^{(3)}_{d u_3 u_4} x^{u_1} x^{u_2} x^{u_3} x^{u_4} $

$\theta^{(n)}_{,c} \theta^{(n)}_{,d} = \Sigma_{j=2}^{2n} \Sigma_{i=1}^{j-i} \tilde{g}^{(i)}_{c \underbrace{u_1 ...}} \tilde{g}^{(j-i)}_{d \underbrace{... u_{j-2}}} x^{u_1} \cdot ... \cdot x^{u_{j-2}}$

first derivative of $g$
$g^{(n)}_{ab,c} = i \theta^{(n)}_{,c} g^{(n)}_{ab}$
$= i (\Sigma_{k=1}^n k \tilde{g}^{(k)}_{c \underbrace{u_1 ... u_{k-1}}} x^{u_1} \cdot ... \cdot x^{u_{k-1}}) g^{(n)}_{ab}$

second derivative of $g$
$g^{(n)}_{ab,cd} = (i \theta^{(n)}_{,c} g^{(n)}_{ab})_{,d}$
$= (i \theta^{(n)}_{,cd} - \theta^{(n)}_{,c} \theta^{(n)}_{,d}) g^{(n)}_{ab}$
$= ( i (\Sigma^{n-2}_{i=1} \tilde{g}^{(i)}_{c d u_1 ... u_i} x^{u_1} \cdot ... \cdot x^{u_i} ) - (\Sigma_{j=2}^{2n} \Sigma_{i=1}^{j-i} \tilde{g}^{(i)}_{c \underbrace{u_1 ...}} \tilde{g}^{(j-i)}_{d \underbrace{... u_{j-2}}} x^{u_1} \cdot ... \cdot x^{u_{j-2}} ) ) g^{(n)}_{ab}$

connection
$\Gamma_{abc} = \frac{1}{2} (g_{ab,c} + g_{ac,b} - g_{bc,a})$
$= i \frac{1}{2} (g_{ab} \theta_{,c} + g_{ac} \theta_{,b} - g_{bc} \theta_{,a})$

${\Gamma^a}_{bc} = i \frac{1}{2} (\delta^a_b \theta_{,c} + \delta^a_c \theta_{,b} - g^{ad} g_{bc} \theta_{,d})$
$= i \frac{1}{2} (\delta^a_b \delta^u_c + \delta^a_c \delta^u_b - g^{au} g_{bc}) \theta_{,u}$

$\Gamma^{(n)}_{abc} = \frac{1}{2} ( g^{(n)}_{ab,c} + g^{(n)}_{ac,b} - g^{(n)}_{bc,a} )$
$= \frac{1}{2} i ( (\Sigma_{k=1}^n k \tilde{g}^{(k)}_{c \underbrace{u_1 ... u_{k-1}}} x^{u_1} \cdot ... \cdot x^{u_{k-1}}) g^{(n)}_{ab} + (\Sigma_{k=1}^n k \tilde{g}^{(k)}_{b \underbrace{u_1 ... u_{k-1}}} x^{u_1} \cdot ... \cdot x^{u_{k-1}}) g^{(n)}_{ac} - (\Sigma_{k=1}^n k \tilde{g}^{(k)}_{a \underbrace{u_1 ... u_{k-1}}} x^{u_1} \cdot ... \cdot x^{u_{k-1}}) g^{(n)}_{bc} )$
$= \frac{1}{2} i (g^{(n)}_{ab} \delta^{u_0}_c + g^{(n)}_{ac} \delta^{u_0}_b - \delta^{u_0}_a g^{(n)}_{bc}) \cdot (\Sigma_{k=1}^n k \tilde{g}^{(k)}_{u_0 \underbrace{u_1 ... u_{k-1}}} x^{u_1} \cdot ... \cdot x^{u_{k-1}})$

${\Gamma^{(n)a}}_{bc} = \frac{1}{2} i g^{(n)ad} (g^{(n)}_{db} \delta^{u_0}_c + g^{(n)}_{dc} \delta^{u_0}_b - \delta^{u_0}_d g^{(n)}_{bc}) \cdot (\Sigma_{k=1}^n k \tilde{g}^{(k)}_{u_0 \underbrace{u_1 ... u_{k-1}}} x^{u_1} \cdot ... \cdot x^{u_{k-1}}) $
$= \frac{1}{2} i (\delta^a_b \delta^{u_0}_c + \delta^a_c \delta^{u_0}_b - g^{(n)a {u_0}} g^{(n)}_{bc}) \cdot (\Sigma_{k=1}^n k \tilde{g}^{(k)}_{u_0 \underbrace{u_1 ... u_{k-1}}} x^{u_1} \cdot ... \cdot x^{u_{k-1}}) $

connection partial
${\Gamma^a}_{bc,d} = (\frac{1}{2} g^{ae} (g_{eb,c} + g_{ec,b} - g_{bc,e}))_{,d}$
$= (\frac{1}{2} g^{ae} (g_{eb,c} + g_{ec,b} - g_{bc,e}))_{,d}$
$= {g^{ae}}_{,d} \Gamma_{ebc} + \frac{1}{2} g^{ae} (g_{eb,cd} + g_{ec,bd} - g_{bc,ed})$
$= -g^{ae} g_{ef,d} {\Gamma^f}_{bc} + \frac{1}{2} g^{ae} (g_{eb,cd} + g_{ec,bd} - g_{bc,ed})$
$= \frac{1}{2} g^{ae} (-2g_{ef,d} {\Gamma^f}_{bc} + g_{eb,cd} + g_{ec,bd} - g_{bc,ed})$

$= \frac{1}{2} g^{ae} (-2 i g_{ef} \theta_{,d} {\Gamma^f}_{bc} + (i \theta_{,cd} - \theta_{,c} \theta_{,d}) g_{eb} + (i \theta_{,bd} - \theta_{,b} \theta_{,d}) g_{ec} - (i \theta_{,ed} - \theta_{,e} \theta_{,d}) g_{bc})$
$= \frac{1}{2} g^{ae} ( g_{ef} \theta_{,d} (\delta^f_b \delta^u_c + \delta^f_c \delta^u_b - g^{fu} g_{bc}) \theta_{,u} + (i \theta_{,fd} - \theta_{,f} \theta_{,d})(g_{eb} \delta^f_c + g_{ec} \delta^f_b - g_{bc} \delta^f_e) )$
$= \frac{1}{2} ( \theta_{,d} \theta_{,u} (\delta^a_b \delta^u_c + \delta^a_c \delta^u_b - g_{bc} g^{au}) + (i \theta_{,fd} - \theta_{,f} \theta_{,d})(\delta^a_b \delta^f_c + \delta^a_c \delta^f_b - g_{bc} g^{af}) )$
$= \frac{1}{2} (\theta_{,d} \theta_{,e} + i \theta_{,ed} - \theta_{,e} \theta_{,d}) (\delta^a_b \delta^e_c + \delta^a_c \delta^e_b - g_{bc} g^{ae}) $
$= \frac{1}{2} i \theta_{,de} (\delta^a_b \delta^e_c + \delta^a_c \delta^e_b - g_{bc} g^{ae}) $

${\Gamma^{(n)a}}_{bc,d} = i \frac{1}{2} (\delta^a_b \delta^e_c + \delta^a_c \delta^e_b - g^{(n)ae} g^{(n)}_{bc}) \Sigma^{n-2}_{k=1} \tilde{g}^{(k)}_{d e u_1 ... u_k} x^{u_1} \cdot ... \cdot x^{u_k}$

Riemann
${R^a}_{bcd} = {\Gamma^a}_{bd,c} - {\Gamma^a}_{bc,d} + {\Gamma^a}_{uc} {\Gamma^u}_{bd} - {\Gamma^a}_{ud} {\Gamma^u}_{bc}$
$= \frac{1}{2} i \theta_{,ce} (\delta^a_b \delta^e_d + \delta^a_d \delta^e_b - g_{bd} g^{ae}) - \frac{1}{2} i \theta_{,de} (\delta^a_b \delta^e_c + \delta^a_c \delta^e_b - g_{bc} g^{ae}) + ( i \frac{1}{2} (\delta^a_u \delta^e_c + \delta^a_c \delta^e_u - g^{ae} g_{uc}) \theta_{,e} ) ( i \frac{1}{2} (\delta^u_b \delta^f_d + \delta^u_d \delta^f_b - g^{uf} g_{bd}) \theta_{,f} ) - ( i \frac{1}{2} (\delta^a_u \delta^e_d + \delta^a_d \delta^e_u - g^{ae} g_{ud}) \theta_{,e} ) ( i \frac{1}{2} (\delta^u_b \delta^f_c + \delta^u_c \delta^f_b - g^{uf} g_{bc}) \theta_{,f} ) $
$= \frac{1}{2} i \theta_{,ce} (\delta^a_b \delta^e_d + \delta^a_d \delta^e_b - g_{bd} g^{ae}) - \frac{1}{2} i \theta_{,de} (\delta^a_b \delta^e_c + \delta^a_c \delta^e_b - g_{bc} g^{ae}) - \frac{1}{4} (\delta^a_u \delta^e_c + \delta^a_c \delta^e_u - g^{ae} g_{uc}) (\delta^u_b \delta^f_d + \delta^u_d \delta^f_b - g^{uf} g_{bd}) \theta_{,e} \theta_{,f} + \frac{1}{4} (\delta^a_u \delta^e_d + \delta^a_d \delta^e_u - g^{ae} g_{ud}) (\delta^u_b \delta^f_c + \delta^u_c \delta^f_b - g^{uf} g_{bc}) \theta_{,e} \theta_{,f} $