x[u_,v_]=(u^2-v^2)/2  
y[u_,v_]=u v
u[x_,y_]=y/Sqrt[-x+Sqrt[x^2+y^2]] 
v[x_,y_]=Sqrt[-x+Sqrt[x^2+y^2]]
Simplify[{D[u[x,y],x],D[u[x,y],y]}]
% /. {x->x[u,v],y->y[u,v]}
eU=FullSimplify[%,{Element[u,Reals],v>0}]
             u        v
Out[7] = {-------, -------}
           2    2   2    2
          u  + v   u  + v

Simplify[{D[v[x,y],x],D[v[x,y],y]}]
% /. {x->x[u,v],y->y[u,v]}
eV=FullSimplify[%,{Element[u,Reals],v>0}]
              v         u
Out[10]= {-(-------), -------}
             2    2    2    2
            u  + v    u  + v

eu=Simplify[{D[x[u,v],u],D[y[u,v],u]}]
Out[11]= {u, v}

ev=Simplify[{D[x[u,v],v],D[y[u,v],v]}]
Out[12]= {-v, u}

gij={{eu.eu,eu.ev},{ev.eu,ev.ev}}
            2    2           2    2
Out[14]= {{u  + v , 0}, {0, u  + v }}

m=Inverse[Transpose[{eu,ev}]] 
              u        v            v         u
Out[14]= {{-------, -------}, {-(-------), -------}}
            2    2   2    2       2    2    2    2
           u  + v   u  + v       u  + v    u  + v

m.D[eu,u]                                                                  
             u          v
Out[15]= {-------, -(-------)}
           2    2     2    2
          u  + v     u  + v

m.D[eu,v]
             v        u
Out[16]= {-------, -------}
           2    2   2    2
          u  + v   u  + v

m.D[ev,u]
            v        u
Out[17]= {-------, -------}
           2    2   2    2
          u  + v   u  + v

m.D[ev,v]
              u         v
Out[18]= {-(-------), -------}
             2    2    2    2
            u  + v    u  + v

conclude:

{u/uu}=u/2r {v/uu}=-v/2r
{u/uv}=v/2r {v/uv}=u/2r
{u/vv}=-u/2r, {v/vv}=v/2r

equ=u''[t] + u[t]/(u[t]^2+v[t]^2) (u'[t]^2 -v'[t]^2) + 2 v[t]/(u[t]^2+v[t]^2) u'[t] v'[t] 

eqv=v''[t] + v[t]/(u[t]^2+v[t]^2) (v'[t]^2 -u'[t]^2) + 2 u[t]/(u[t]^2+v[t]^2) u'[t] v'[t] 

{x0,y0,dx0,dy0}={1.5,-.25,-1,1}
u0=u[x0,y0]
v0=v[x0,y0]
du0=(u0 dx0+ v0 dy0)/(u0^2+v0^2)
dv0=(u0 dy0- v0 dx0)/(u0^2+v0^2)

solution =NDSolve[{equ==0,eqv==0,u[0]==u0,v[0]==v0,u'[0]==du0,v'[0]==dv0},{u,v},{t,0,5}]

ParametricPlot[Evaluate[{u[t],v[t]} /. solution],{t,0,5},AspectRatio->Automatic]

ParametricPlot[Evaluate[{(u[t]^2-v[t]^2)/2,u[t] v[t]} /. solution],{t,0,1},AspectRatio->Automatic]

gijk=ConstantArray[0,{2,2,2}];
Do[ Do[ Do[
gijk[[i,j,k]]=If[k==1,D[gij[[i,j]],u],D[gij[[i,j]],v]],{k,2}],{j,2}],{i,2}]

gIJ=Inverse[gij]
Glik=ConstantArray[0,{2,2,2}];
Do[ Do[ Do[
sum=Sum[gIJ[[l,j]] (gijk[[i,j,k]]+gijk[[j,k,i]]-gijk[[k,i,j]]),{j,2}];
Glik[[l,i,k]]=sum/2,
{k,2}],{l,2}],{i,2}]

Glik[[2,All,All]]                                                      
                v         u          u        v
Out[35]= {{-(-------), -------}, {-------, -------}}
              2    2    2    2     2    2   2    2
             u  + v    u  + v     u  + v   u  + v

Glik[[1,All,All]]                                                      
              u        v          v          u
Out[36]= {{-------, -------}, {-------, -(-------)}}
            2    2   2    2     2    2     2    2
           u  + v   u  + v     u  + v     u  + v


---
Glijk=ConstantArray[0,{2,2,2,2}];
Do[ Do[ Do[ Do [
Glijk[[l,i,j,k]]=Switch[k,1,D[Glik[[l,i,j]],u],2,D[Glik[[l,i,j]],v]],{k,2}],{l,2}],{j,2}],{i,2}]

Rdabc=ConstantArray[0,{2,2,2,2}];

Do[ Do[ Do[ Do [
sum1=Sum[Glik[[j,a,b]] Glik[[d,j,c]]-Glik[[j,a,c]] Glik[[d,j,b]],{j,2}];
Rdabc[[d,a,b,c]]=FullSimplify[sum1+Glijk[[d,a,b,c]]-Glijk[[d,a,c,b]] ],{a,2}],{b,2}],{c,2}],{d,2}]

