Let $ mathcal {S} $ to be the symmetric matrices and $ mathcal {P} $ to be positive defined matrices.

$ mathcal {S} $ naturally carries the structure of a vector space. Indoor product on $ mathcal {S} $ is given by $ langle A, B rangle = trace (A B) $ .

$ mathcal {P} $ is an open set in ( mathcal {S} $.

the map $$ log: mathcal {P} rightarrow mathcal {S} $$

is a diffeomorphism between two varieties. We can identify the tangent space in x $ T _ { text {x}} mathcal {P} $ with $ text {x} times mathcal {S} $.

The metric induced on the Tangent space is given by $$ langle A, B rangle_ {x} = langle dlog (A) | _ {x}, dlog (B) | _ {x} rangle $$ ,or

$$

dlog: T_x mathcal {P} rightarrow T_x mathcal {S} $$

$$

An mapsto an X ^ {- 1}

$$

Explicitly write the inner product on $ mathcal {P} $ is given by

$$ langle A, B rangle_ {X} = text {trace} (A X ^ {- 1} B X ^ {- 1}) $$

Length $ L ( gamma) $ and energy $ E ( gamma) $ of a curve $ gamma: [0,1] rightarrow mathcal {P} $ is given by

$$

L ( gamma) = int_a ^ b | dfrac {d gamma} {dt} | _ { gamma (t)} dt $$

$$ E ( gamma) = frac {1} {2} int_a ^ b | dfrac {d gamma} {dt} | _ { gamma (t)} ^ 2 dt $$

Geodesics are curves that minimize energy on a collector. They allow us to introduce a distance between two points of a manifold. To calculate them we can simply use euler's lagler equations

$$ frac {d} {d}} frac {d} {d dot { gamma}} f (t, gamma (t), dot { gamma} (t)) = frac {d} {d gamma} f (t, gamma (t), point { gamma} (t)) $$

write explicitly we have

$$ frac {d} {dt} frac {d} {d dot { gamma}}

text {trace} ( dot { gamma} gamma ^ {- 1} dot { gamma} gamma ^ {- 1}) = frac {d} {d gamma}

text {trace} ( dot { gamma} gamma ^ {- 1} dot { gamma} gamma ^ {- 1}) $$

The left side of the equation reduces to

$$ frac {d} {d}} gamma ^ {- 1} dot { gamma} gamma ^ {- 1} $$

while the right side of the equation reduces to

$$ – gamma ^ {- 1} dot { gamma} gamma ^ {- 1} dot { gamma} gamma ^ {- 1} $$

By solving the differential equation, we obtain two different expressions for geodesics.

$$ gamma (t) = P exp (t P ^ {- 1} S) $$

Here, P is located on the variety and S is located on the tangent space. intuitively $ gamma $ is the curve that starts in P with the direction S.

$$ gamma_ {AB} (t) = A (A ^ {- 1} B) ^ t $$

Here, gamma is the geodesic between the points of variety A and B.

Now that the geodesics are defined, the distance on the spd manifold can be defined via the length of the geodesic curve connecting two points. After some calculations we come to

$$ d (X, Y) = | log (X ^ {- 1} Y) | $$

So, these are my calculations up to now. I am pretty new to differential geometry. My first question is: Do arguments usually make sense? I'm not really convinced that $ dlog = dX X ^ {- 1} $ or that the internal product on $ mathcal {S} $ is $ trace (AB) $. My ultimate goal is to read the christoffel symbols from the geodesic eq. and calculate the riem. curvature. But I'm not sure how to proceed or if calculations have up to now a meaning