Let $\mathcal{S}$ be the Symmetric Matrices and $\mathcal{P}$ be the positive definite matrices. $\mathcal{S}$ naturally carries the structure of a vector space. Inner 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 manifolds. We can identify the tangent space at x $ T_{\text{x}}\mathcal{P} $ with $\text{x} \times \mathcal{S} $. Induced metric on the Tangent space is given by $$ \langle A , B \rangle_{x} = \langle dlog(A) |_{x} , dlog(B) |_{x} \rangle $$ ,where $$ dlog : T_x\mathcal{P} \rightarrow T_{log(x)}\mathcal{S} $$ $$ A \mapsto A X^{-1} $$ Writing out explicitly 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 energy minimizing curves on a manifold. They allow us to introduce distance between two points on a manifold. To calculate them we can simply employ euler lagrange equations
$$ \frac{d}{dt} \frac{d}{d\dot{\gamma}} f(t,\gamma(t), \dot{\gamma}(t)) = \frac{d}{d\gamma} f(t,\gamma(t), \dot{\gamma}(t)) $$
writing out explicitly we have
$$ \frac{d}{dt} \frac{d}{d\dot{\gamma}} \text{trace}( \dot{\gamma} \gamma^{-1} \dot{\gamma} \gamma ^{-1}) = \frac{d}{d\gamma} \text{trace}( \dot{\gamma} \gamma^{-1} \dot{\gamma} \gamma ^{-1}) $$
Lefthand side of the equation reduces to
$$ \frac{d}{dt} \gamma^{-1} \dot{\gamma} \gamma ^{-1} $$
whereas the righthand side of the equation reduces to
$$ - \gamma^{-1} \dot{\gamma} \gamma^{-1} \dot{\gamma} \gamma^{-1} $$
Solving the differential equation we get two different expressions for the geodesics. $$ \gamma(t) = P exp(t P^{-1} S) $$ here P lies on the manifold and S lies on the tangent space. Intuitively $\gamma$ is the curve, which starts at P with direction S. $$ \gamma_{AB}(t) = A( A^{-1} B)^t $$ here gamma is the geodesic between the points on the manifold A and B.
Now that the geodesics are defined, distance on the spd manifold may be defined via the length of the geodesic curve connecting two points. After some calculations we arrive at $$ d(X,Y) = \| log( X^{-1} Y ) \| $$
So those are my calculations so far. I am quite new to the differential geometry. My first question is do the arguments generally make sense? I am not really convinced that $ dlog = dX X^{-1} $ or that the inner product on $\mathcal{S}$ is $ trace(AB)$. My ultimate goal is to read christoffel symbols from the geodesics eq. and calculate the riem. curvature. But i am not sure how to proceed or if the calculations so far makes sense.
Edit: I tried to find another expression for dlog (and hoped it would be equivalent to $ AX^{-1}$) by taking the directional derivative of log-series . But generally the direction in which i take the derivative and the base point wont commute. That is why i dont think my expression for dlog is true. It is somehow funny that a wrong expression still produces a valid metric.