7 On Geodesic Triangles with Right Angles in a Dually Flat Space
161
geometric entities that can be interpreted as multilinear functionals over Cartesian
products of dual covector and vector spaces. A vector v of T p is a rank-1 tensor
that can be expressed either using the covariant or contravariant components of a
reciprocal basis. Thus one should not confuse the notion of “geometric vectors” that
are tensors (coordinate-free objects independent of the choice of the basis) with the
“column vectors” of vector components in a basis which are used to perform linear algebra calculations on (geometric) vectors. When the components of a tensor
vector varies with the inverse transformation of the change of basis, we say that we
have a contravariant (tensor) vector (a (0, 1)-tensor), and its components are called
contravariant components. When the components of a tensor vector varies according to the transformation of the change of basis, we say that we have a covariant
(tensor) vector (a covector or (1, 0)-tensor, i.e., an element of the dual vector space
V
∗ of linear functionals or linear forms), and its components are called covariant
components. The metric tensor g is a (2, 0) covariant tensor [15].
7.2.2 Dual Geodesics and Their Tangent Vectors, and Dually
Coupled Parallel Transport
Let F(θ ) be a D-dimensional C
3 real-valued function defined on an open convex
domain , and denote by F
∗
(η) its Legendre-Fenchel convex conjugate [3, 15]:
F
∗
(η) = sup θ∈ θ
η − F(θ ). The dual potential functions F and F
∗ induce two
torsion-free flat affine connections [15] ∇ and ∇
∗ , respectively. A Bregman manifold
M is equipped with two global affine coordinate systems θ(·) (the ∇-affine coordinate
system) and η(·) (the ∇
∗ -affine coordinate system) such that it comes from LegendreFenchel transformation that η(θ) = ∇ F(θ ) and θ(η) = ∇ F
∗
(η). Let θ
i
( p) and η i ( p)
denote the primal i-th θ -coordinate functions and the dual i-th η-coordinate functions
of a point p, for i ∈ {1, . . . , D} so that θ( p) = (θ
1
( p), . . . , θ
D
( p)) and η( p) =
(η 1 ( p), . . . , η D ( p)). Notations are summarized in Appendix A. Any point p ∈ M
can be expressed equivalently either in the primal global θ -chart or the dual global
η-chart. The dual geodesics
3
γ pq and γ
∗
pq passing through two given points p, q ∈ M
write simply using the dual coordinate systems as follows:
γ pq = {x λ ∈ M : θ(x λ ) = (1 − λ)θ ( p) + λθ (q), λ ∈ [0, 1]},
(7.9)
γ
∗
pq = {x λ ∈ M : η(x λ ) = (1 − λ)η( p) + λη(q), λ ∈ [0, 1]}.
(7.10)
In general, a vector field v(t) is parallel along a smooth curve c(t) iff.
∀i ∈ {1, . . . , D}, ˙
v
i
+
D
j,k=1
i
jk ˙
x
j
v
k
= 0.
(7.11)
3 Given an affine connection ∇, the ∇-geodesic is an autoparallel curve [15].
161
geometric entities that can be interpreted as multilinear functionals over Cartesian
products of dual covector and vector spaces. A vector v of T p is a rank-1 tensor
that can be expressed either using the covariant or contravariant components of a
reciprocal basis. Thus one should not confuse the notion of “geometric vectors” that
are tensors (coordinate-free objects independent of the choice of the basis) with the
“column vectors” of vector components in a basis which are used to perform linear algebra calculations on (geometric) vectors. When the components of a tensor
vector varies with the inverse transformation of the change of basis, we say that we
have a contravariant (tensor) vector (a (0, 1)-tensor), and its components are called
contravariant components. When the components of a tensor vector varies according to the transformation of the change of basis, we say that we have a covariant
(tensor) vector (a covector or (1, 0)-tensor, i.e., an element of the dual vector space
V
∗ of linear functionals or linear forms), and its components are called covariant
components. The metric tensor g is a (2, 0) covariant tensor [15].
7.2.2 Dual Geodesics and Their Tangent Vectors, and Dually
Coupled Parallel Transport
Let F(θ ) be a D-dimensional C
3 real-valued function defined on an open convex
domain , and denote by F
∗
(η) its Legendre-Fenchel convex conjugate [3, 15]:
F
∗
(η) = sup θ∈ θ
η − F(θ ). The dual potential functions F and F
∗ induce two
torsion-free flat affine connections [15] ∇ and ∇
∗ , respectively. A Bregman manifold
M is equipped with two global affine coordinate systems θ(·) (the ∇-affine coordinate
system) and η(·) (the ∇
∗ -affine coordinate system) such that it comes from LegendreFenchel transformation that η(θ) = ∇ F(θ ) and θ(η) = ∇ F
∗
(η). Let θ
i
( p) and η i ( p)
denote the primal i-th θ -coordinate functions and the dual i-th η-coordinate functions
of a point p, for i ∈ {1, . . . , D} so that θ( p) = (θ
1
( p), . . . , θ
D
( p)) and η( p) =
(η 1 ( p), . . . , η D ( p)). Notations are summarized in Appendix A. Any point p ∈ M
can be expressed equivalently either in the primal global θ -chart or the dual global
η-chart. The dual geodesics
3
γ pq and γ
∗
pq passing through two given points p, q ∈ M
write simply using the dual coordinate systems as follows:
γ pq = {x λ ∈ M : θ(x λ ) = (1 − λ)θ ( p) + λθ (q), λ ∈ [0, 1]},
(7.9)
γ
∗
pq = {x λ ∈ M : η(x λ ) = (1 − λ)η( p) + λη(q), λ ∈ [0, 1]}.
(7.10)
In general, a vector field v(t) is parallel along a smooth curve c(t) iff.
∀i ∈ {1, . . . , D}, ˙
v
i
+
D
j,k=1
i
jk ˙
x
j
v
k
= 0.
(7.11)
3 Given an affine connection ∇, the ∇-geodesic is an autoparallel curve [15].