Bivariate Gaussian

(1)affine immersion :

We can define an affine immersion  from  the bivariate normal manifold (with zero means and identical  variance σ ) to (R^3)   by a graph immerion:  f: ( (Subscript[θ, 1]),(Subscript[θ, 2]))-> ( (Subscript[θ, 1]),(Subscript[θ, 2]),ψ) 
where  ( (Subscript[θ, 1])=-(σ/(2( (σ^2)-((Subscript[σ, 12])^2) ))) and  (Subscript[θ, 2])=((Subscript[σ, 12])/(( (σ^2)-((Subscript[σ, 12])^2) ))) so here we have (Subscript[θ, 1])<0 , (Subscript[θ, 2])>0 and 4 ((Subscript[θ, 1])^2)>((Subscript[θ, 2])^2))

So   S   is the affine embedding of the bivariate normal manifold  as a surface in (R^3) : where  a= (Subscript[θ, 1])  and  b=(Subscript[θ, 2])

AffFBN(a_,b_):={a,b,log( ((2 π)/(Sqrt[4 (a^2)-(b^2)])) )}

S=ParametricPlot3D[Evaluate[AffFBN(a,b)],{a,-4,-1.001},{b,-1,1},ViewPoint->{3,2,1},BoxRatios->{1,1,1}];

The independent  case  ( b=(Subscript[θ, 2])=0) is represented by this curve M:[so every point in this curve reperesent  bivariate distribution which 
has two normal marginal functions with  zero mean and identical variance ]

AffF2independent(a_):={a,0,log( (( π)/(Sqrt[ (a^2)])) )}

M=ParametricPlot3D[Evaluate[AffF2independent(a)],{a,-4,-1.001},ViewPoint->{3,2,1},BoxRatios->{1,1,1}];

So here we can  construct a neighbouhood  around  this curve which reperesent the  independent case ; as follows :

c[t_]:={t,0,log( (( π)/(Sqrt[(t^2)])) )}

tubecurve[c][r][t,theta]

tubecurve[c][r_][t_,theta_]:={t-((r Cos[theta])/((Sqrt[1+(1/(t^2))]) (Sqrt[(1/(t^4))]) (t^3))),0-((r Sin[theta])/((Sqrt[(1/(t^4))]) (t^2))),log( (( π)/(Sqrt[(t^2)])) )-((r Cos[theta])/((Sqrt[1+(1/(t^2))]) (Sqrt[(1/(t^4))]) (t^2)))};

R=ParametricPlot3D[tubecurve[c][0.6][a,theta]//Evaluate,{a,-4,-1.001},{theta,0,2Pi},PlotPoints->{20,20},ViewPoint->{3,2,1},BoxRatios->{1,1,1}];

This  neighbouhood will inersect the affine embedding  S  of the manifold of bivariate normal distributions :

Show[S,R];

(2)Continuous image of the  affine immersion f :

Continuous image of the affie immersion  f  as a surface in (R^3) using standard coordinates  (σ,(Subscript[σ, 12]))for the bivariate 
normal manifold (with zero means and identical  variance σ ): where  a= σ  and  b=(Subscript[σ, 12]) and   (a^2)>(b^2)

AffFBN(a_,b_):={a,b,log( 2 π(Sqrt[ (a^2)-(b^2)]) )}

S=ParametricPlot3D[Evaluate[AffFBN(a,b)],{a,1.001,4},{b,-1,1},ViewPoint->{3,2,1},BoxRatios->{1,1,1}];

AffF2independent(a_):={a,0,log( 2 π a )}

M=ParametricPlot3D[Evaluate[AffF2independent(a)],{a,1.001,4},ViewPoint->{3,2,1},BoxRatios->{1,1,1}];

So here we can  construct a neighbouhood  around  the curve M which reperesent the  independent case ((Subscript[σ, 12])=0); as follows :

c[t_]:={t,0,log(2 πt)}

tubecurve[c][r][t,theta]

tubecurve[c][r_][t_,theta_]:={t-((r Cos[theta])/((Sqrt[1+(1/(t^2))]) (Sqrt[(1/(t^4))]) (t^3))),0-((r Sin[theta])/((Sqrt[(1/(t^4))]) (t^2))),log(2 πt)-((r Cos[theta])/((Sqrt[1+(1/(t^2))]) (Sqrt[(1/(t^4))]) (t^2)))};

R=ParametricPlot3D[tubecurve[c][0.6][a,theta]//Evaluate,{a,1.001,4},{theta,0,2Pi},PlotPoints->{20,20},ViewPoint->{3,2,1},BoxRatios->{1,1,1}];

This neighbouhood  around  the curve  (Subscript[σ, 12])=0 intersects with bivariate normal manifold (with zero means and identical variance σ )to yield the neighbouhood of bivariate normal distributions:

Show[S,R];

here with natural parameters:???

AffFBN(a_,b_):={a,b,log( ((2 π)/(Sqrt[4 (a^2)-(b^2)])) )}

S=ParametricPlot3D[Evaluate[AffFBN(a,b)],{a,((-4)/32),((-1.001)/32)},{b,((-1)/16),(1/16)},ViewPoint->{3,2,1},BoxRatios->{1,1,1}];