(1)Univariate gamma distributions with respect to the natural coordinates(β,ν):

f=(((β^ν) (E^(-β x)) (x^(ν-1)))/(Γ(ν)));

TeXForm[(((β^ν) (E^(-β x)) (x^(ν-1)))/(Γ(ν)))]

Dimension  M:

n=2 ;

FullSimplify[(∫_0^∞)f  \[DifferentialD]x,{β>0,ν>0}]

FullSimplify[(∫_0^∞)x f  \[DifferentialD]x,{β>0,ν>0}]

FullSimplify[(∫_0^∞)(x^2) f  \[DifferentialD]x,{β>0,ν>0} ]- ((((ν/β)))^2)//FullSimplify

Fisher information matrix g:

ψ=log(Γ(ν))-ν log(β);

(∂_ν) ψ

g=Outer[FullSimplify[Simplify[(((∂^2)ψ)/(∂#1 ∂#2))],{β>0,ν>0}]&,{β,ν},{β,ν}]

The (α)-geometry:

1-The components (Subscript[T, ijk]):

T=Outer[FullSimplify[(((∂^3)ψ)/(∂#1 ∂#2 ∂#3)),{β>0,ν>0}]&,{β,ν},{β,ν},{β,ν}]

1-The components (Subscript[Γ, ij,k]):

(Overscript[Γ, α])=Table[(1/2) (1-α) FullSimplify[T[[i,j,k]]],{i,n},{j,n},{k,n}]

1-The components (Overscript[(Subscript[Γ, ij]), k]) of an affine connection (∇^α):

Table[FullSimplify[Solve[{(Overscript[Γ, α])[[i,j,1]]==(∑_(h=1)^n)g[[1,h]] (Overscript[Γ, h])(i,j),(Overscript[Γ, α])[[i,j,2]]==(∑_(h=1)^n)g[[2,h]] (Overscript[Γ, h])(i,j)},{(Overscript[Γ, 1])(i,j),(Overscript[Γ, 2])(i,j)}]],{i,n},{j,n}]

3-Curvature tensor:

R=FullSimplify[Table[(1/4) (1-(α^2)) Simplify[(∑_(p=1)^n)(∑_(m=1)^n)(T[[k,m,i]] T[[j,l,p]]-T[[k,m,j]] T[[i,l,p]])],{i,n},{j,n},{k,n},{l,n}]]

4-Sectional curvature:

ϱ=FullSimplify[Table[If[i==j,0,-((R[[i,j,i,j]])/(g[[i,i]] g[[j,j]]-((g[[i,j]])^2)))],{i,n},{j,n}]]

5-Mean curvature:

(OverBar[ϱ])=FullSimplify[Table[(∑_(j=1)^n) ϱ[[i,j]],{i,n}]]

6-Ricci curvature:

R=FullSimplify[Table[(∑_(l=1)^n)(∑_(i=1)^n)(g^(-1))[[i,l]] R[[i,j,k,l]],{j,n},{k,n}]]

7-Scalar curvature:

R=FullSimplify[(∑_(j=1)^n)(∑_(i=1)^n)(g^(-1))[[i,j]] R[[i,j]]]

7-Gaussain curvature:

K=(R/2)

The expectation coordiantes ((Subscript[η, i])):

η=Outer[FullSimplify[((∂ψ)/(∂#1)),{β>0,ν>0}]&,{β,ν}]

ϕ=FullSimplify[-ψ+β η[[1]]+ν η[[2]]]

(2)Univariate gamma distributions with respect to the local coordinates(β,ν):

f=((((((ν/β)))^ν) (E^(-x(ν/β))) (x^(-1+ν)))/(Γ(ν)));

Dimension  M:

n=2 ;

FullSimplify[(∫_0^∞)f  \[DifferentialD]x,{β>0,ν>0}]

g=Outer[FullSimplify[(∫_0^∞)-f Simplify[(((∂^2)log(f))/(∂#1 ∂#2))]\[DifferentialD]x,{β>0,ν>0}]&,{β,ν},{β,ν}]

The (α)-geometry:

(Subscript[α, 1])=β;

(Subscript[α, 2])=ν;

1-The components (Subscript[Γ, ij,k]) :

(Overscript[Γ, α])=Table[FullSimplify[(∫_0^∞)f  Simplify[(1/2) (1-α) ((∂log(f))/(∂(Subscript[α, i]))) ((∂log(f))/(∂(Subscript[α, j])))+(((∂^2)log(f))/(∂(Subscript[α, i]) ∂(Subscript[α, j])))] ((∂log(f))/(∂(Subscript[α, k])))\[DifferentialD]x,{β>0,ν>0}],{i,n},{j,n},{k,n}]

2-The components (Subscript[(Overscript[Γ, k]), ij]) of an affine connection (∇^α):

Table[FullSimplify[Solve[(Overscript[Γ, α])[[i,j,1]]==(∑_(h=1)^n)g[[1,h]] (Overscript[Γ, h])[i,j]&&(Overscript[Γ, α])[[i,j,2]]==(∑_(h=1)^n)g[[2,h]] (Overscript[Γ, h])[i,j],{(Overscript[Γ, 1])[i,j],(Overscript[Γ, 2])[i,j]}]],{i,n},{j,n}]

(Overscript[Γ, 1])=(({{(1/2) (((3 α-1)/((Subscript[α, 1])+(Subscript[α, 3])))-((α+1)/(Subscript[α, 1]))), 0, ((α-1)/(2 ((Subscript[α, 1])+(Subscript[α, 3])))), 0}, {0, (((α+1) (Subscript[α, 1]) (Subscript[α, 3]))/(2 (Subscript[α, 2]^2) ((Subscript[α, 1])+(Subscript[α, 3])))), 0, 0}, {((α-1)/(2 ((Subscript[α, 1])+(Subscript[α, 3])))), 0, (((α+1) (Subscript[α, 1]))/(2 (Subscript[α, 3]) ((Subscript[α, 1])+(Subscript[α, 3])))), 0}, {0, 0, 0, -(((α+1) (Subscript[α, 1]) (Subscript[α, 3]))/(2 ((Subscript[α, 1])+(Subscript[α, 3])) (Subscript[α, 4]^2)))}}));

(Overscript[Γ, 2])=(({{0, ((α-1)/(2 ((Subscript[α, 1])+(Subscript[α, 3])))), 0, 0}, {((α-1)/(2 ((Subscript[α, 1])+(Subscript[α, 3])))), ((α-1)/(Subscript[α, 2])), -(((α-1) (Subscript[α, 1]))/(2 (Subscript[α, 3]) ((Subscript[α, 1])+(Subscript[α, 3])))), 0}, {0, -(((α-1) (Subscript[α, 1]))/(2 (Subscript[α, 3]) ((Subscript[α, 1])+(Subscript[α, 3])))), 0, 0}, {0, 0, 0, 0}}));

3-Curvature tensor:

R=FullSimplify[Table[(∑_(h=1)^n)g[[h,l]] Simplify[-((∂(Overscript[Γ, h])[[i,k]])/(∂(Subscript[α, j])))+((∂(Overscript[Γ, h])[[j,k]])/(∂(Subscript[α, i])))+(∑_(m=1)^n)Simplify[(Overscript[Γ, h])[[i,m]] (Overscript[Γ, m])[[j,k]]-(Overscript[Γ, h])[[j,m]] (Overscript[Γ, m])[[i,k]]]],{i,n},{j,n},{k,n},{l,n}]]

4-Sectional curvature:

ϱ=FullSimplify[Table[If[i==j,0,-((R[[i,j,i,j]])/(g[[i,i]] g[[j,j]]-((g[[i,j]])^2)))],{i,n},{j,n}]]

5-Mean curvature:

(OverBar[ϱ])=FullSimplify[Table[(∑_(j=1)^n)(1/3) ϱ[[i,j]],{i,n}]]

6-Ricci curvature:

R=FullSimplify[Table[(∑_(l=1)^n)(∑_(i=1)^n)(g^(-1))[[i,l]] R[[i,j,k,l]],{j,n},{k,n}]]

7-Scalar curvature:

R=FullSimplify[(∑_(j=1)^n)(∑_(i=1)^n)(g^(-1))[[i,j]] R[[i,j]]]

7-Gaussain curvature:

K=(R/2)