Gaussian Manifold and Geodesics

Normal Distributions: N(μ,σ)

The Normal probability density function is given by  N(μ,σ)=(1/((Sqrt[2π])σ))(e^((-(((x-μ))^2))/(2 (σ^2))))  , where μ  is the mean,  and  σ is the  standard deviation.  The Fisher metric for the  Normal  Manifold  NM with respect to the coordinate system  (μ,σ)  is given by  
g=(({{(1/(σ^2)), 0}, {0, (2/(σ^2))}})),and the Christoffel  symbols are given by:
((Subscript[Γ, 11])^1)=0;
             ((Subscript[Γ, 12])^1)=((Subscript[Γ, 21])^1)=((-1)/σ);
             ((Subscript[Γ, 22])^1)=0;
             ((Subscript[Γ, 11])^2)=(1/(2σ));
             ((Subscript[Γ, 12])^2)=((Subscript[Γ, 21])^2)=0;
             ((Subscript[Γ, 22])^2)=((-1)/σ).

Finding Geodesics Numerically for the Normal Manifold:

A geodesic on a manifold M is a curve C(t)=(x(t), y(t))   such that:
x''+((Subscript[Γ, 11])^1) (((x'))^2)+2 ((Subscript[Γ, 12])^1) x'y'+((Subscript[Γ, 22])^1) (((y'))^2)=0   and 
             y''+((Subscript[Γ, 11])^2) (((x'))^2)+2 ((Subscript[Γ, 12])^2) x'y'+((Subscript[Γ, 22])^2) (((y'))^2)=0;  where the ((Subscript[Γ, jk])^i) are Christoffel symboles. 

On the Normal Manifold the geodesic equations are reperesented by:
x''=(2/σ) x'y' ;  and 
        y''=((-1)/(2 σ)) (((x'))^2)+(1/σ) (((y'))^2).   They are independent of the mean μ.
....................................................................................................................

x''[t]==(2/σ)x'[t]*y'[t]
y''[t]==((-1)/(2σ))(((x'[t]))^2)+(1/σ)(((y'[t]))^2)

Solving the geodesics equations for Normal manifold:???

x''(t)==((2 x'(t) y'(t))/σ)
y''(t)==(((y'(t))^2)/σ)-(((x'(t))^2)/(2 σ))

DSolve[{ σ  x''(t)==2 x'(t) y'(t) , σ  y''(t)==((y'(t))^2)-(1/2)((x'(t))^2) }, {x(t) , y(t)}, t]

1)Normal geodesics (x(t),y(t)) where σ=1 centered at the origin:(N(μ,1))

With[{μ =μ, σ=1},normal=Flatten[Table[NDSolve[{x''[t]==(2/σ)x'[t]*y'[t],y''[t]==((-1)/(2σ))(((x'[t]))^2)+(1/σ)(((y'[t]))^2),x[0]==0,y[0]==0,x'[0]==Cos[a],y'[0]==Sin[a]},{x,y},{t,1,10}],{a,0,2Pi,Pi/12}],1]];
NG[μ ,1]=ParametricPlot[Evaluate[{x[t],y[t]}/. normal],{t,0,2.4},AspectRatio->Automatic,Axes->False,PlotRange->{{-2,2},{-1.5,.3}},Prolog->AbsoluteThickness[1]]

Circles for the normal geodesics NG[μ,1]:

With[{μ =μ, σ=1},normal=Flatten[Table[NDSolve[{x''[t]==(2/σ)x'[t]*y'[t],y''[t]==((-1)/(2σ))(((x'[t]))^2)+(1/σ)(((y'[t]))^2),x[0]==0,y[0]==0,x'[0]==Cos[a],y'[0]==Sin[a]},{x,y},{t,1,10}],{a,0,2Pi,Pi/12}],1]];
NG[μ ,1]=ParametricPlot[Evaluate[{x[t],y[t]}/. normal],{t,0,2.4},AspectRatio->Automatic,Axes->False,PlotRange->{{-2,2},{-1.5,.3}},Prolog->AbsoluteThickness[1]];
Do[vc[i]=ListPlot[Evaluate[{x[i/7],y[i/7]}/.normal],Joined->True,AspectRatio->Automatic,Axes->False,PlotRange->{{-2,2},{-1.5,.5}},PlotStyle->{SpecCol[i]},(*include this line for color*)Prolog->AbsoluteThickness[1],DisplayFunction->Identity],{i,14}];
B=Show[vc[1],vc[2],vc[3],vc[4],vc[5],vc[6],vc[7],vc[8],vc[9],DisplayFunction->$DisplayFunction,Axes -> Automatic];
Show[NG[μ ,1], B]

2)Normal geodesics (x(t),y(t)) where σ=3 centered at the origin:(N(μ,3))

With[{μ =μ, σ=3},normal=Flatten[Table[NDSolve[{x''[t]==(2/σ)x'[t]*y'[t],y''[t]==((-1)/(2σ))(((x'[t]))^2)+(1/σ)(((y'[t]))^2),x[0]==0,y[0]==0,x'[0]==Cos[a],y'[0]==Sin[a]},{x,y},{t,1,10}],{a,0,2Pi,Pi/12}],1]];
NG[μ ,3]=ParametricPlot[Evaluate[{x[t],y[t]}/. normal],{t,0,2.4},AspectRatio->Automatic,Axes->False,PlotRange->{{-2,2},{-1.5,1.8}},Prolog->AbsoluteThickness[1]];
Do[vc[i]=ListPlot[Evaluate[{x[i/7],y[i/7]}/.normal],Joined->True,AspectRatio->Automatic,Axes->False,PlotRange->{{-2,2},{-1.5,1.8}},PlotStyle->{SpecCol[i]},(*include this line for color*)Prolog->AbsoluteThickness[1],DisplayFunction->Identity],{i,14}];
B=Show[vc[1],vc[2],vc[3],vc[4],vc[5],vc[6],vc[7],vc[8],vc[9],DisplayFunction->$DisplayFunction,Axes -> Automatic];
Show[NG[μ ,3], B]

3)Normal geodesics (x(t),y(t)) where σ=2 centered at the origin:(N(μ,2))

With[{μ =μ, σ=2},normal=Flatten[Table[NDSolve[{x''[t]==(2/σ)x'[t]*y'[t],y''[t]==((-1)/(2σ))(((x'[t]))^2)+(1/σ)(((y'[t]))^2),x[0]==0,y[0]==0,x'[0]==Cos[a],y'[0]==Sin[a]},{x,y},{t,1,10}],{a,0,2Pi,Pi/4}],1]];
NG[μ ,2]=ParametricPlot[Evaluate[{x[t],y[t]}/. normal],{t,0,2.4},AspectRatio->Automatic,Axes->False,PlotRange->{{-2,2},{-1.5,2.2}},Prolog->AbsoluteThickness[1]];
Do[vc[i]=ListPlot[Evaluate[{x[i/7],y[i/7]}/.normal],Joined->True,AspectRatio->Automatic,Axes->False,PlotRange->{{-2,2},{-1.5,2.2}},PlotStyle->{SpecCol[i]},(*include this line for color*)Prolog->AbsoluteThickness[1],DisplayFunction->Identity],{i,14}];
B=Show[vc[1],vc[2],vc[3],vc[4],vc[5],vc[6],vc[7],vc[8],vc[9],DisplayFunction->$DisplayFunction,Axes -> Automatic];
Show[NG[μ ,2], B]

4)Normal geodesics (x(t),y(t)) where σ=0.5 centered at the origin:(N(μ,.5))

With[{μ =μ, σ=.5},normal=Flatten[Table[NDSolve[{x''[t]==(2/σ)x'[t]*y'[t],y''[t]==((-1)/(2σ))(((x'[t]))^2)+(1/σ)(((y'[t]))^2),x[0]==0,y[0]==0,x'[0]==Cos[a],y'[0]==Sin[a]},{x,y},{t,1,10}],{a,0,2Pi,Pi/12}],1]];
NG[μ ,.5]=ParametricPlot[Evaluate[{x[t],y[t]}/. normal],{t,0,3},AspectRatio->Automatic,Axes->False,PlotRange->{{-2,2},{-1.3,2.5}},Prolog->AbsoluteThickness[1]]

With[{μ =μ, σ=.5},normal=Flatten[Table[NDSolve[{x''[t]==(2/σ)x'[t]*y'[t],y''[t]==((-1)/(2σ))(((x'[t]))^2)+(1/σ)(((y'[t]))^2),x[0]==0,y[0]==0,x'[0]==Cos[a],y'[0]==Sin[a]},{x,y},{t,1,10}],{a,0,2Pi,Pi/12}],1]];
NG[μ ,.5]=ParametricPlot[Evaluate[{x[t],y[t]}/. normal],{t,0,3},AspectRatio->Automatic,Axes->False,PlotRange->{{-2,2},{-1.3,2.5}},Prolog->AbsoluteThickness[1]];
Do[vc[i]=ListPlot[Evaluate[{x[i/7],y[i/7]}/.normal],Joined->True,AspectRatio->Automatic,Axes->False,PlotRange->{{-2,2},{-1.3,2.5}},PlotStyle->{SpecCol[i]},(*include this line for color*)Prolog->AbsoluteThickness[1],DisplayFunction->Identity],{i,14}];
B=Show[vc[1],vc[2],vc[3],vc[4],vc[5],vc[6],vc[7],vc[8],vc[9],DisplayFunction->$DisplayFunction,Axes -> Automatic];
Show[NG[μ ,.5], B]