I'm reading the documentation of this [package: Manopt](https://manoptjl.org/v0.1/manifolds/hyperbolic/), and they claim that in the hyperboloid model for $\mathbb{H}^d$ the parallel transport between tangent spaces $T\_x$ and $T\_y$ is given for any $u\in T\_x$ by
$$
 P\_{x\mapsto y}:\,
 u
 \mapsto 
 u 
 -
 \frac{
 \langle 
 \exp\_{x}^{-1}(y)
 ,
 u
 \rangle\_x
 }{
 d\_{\mathbb{H}^d}^2(x,y)
 }
 \,
 \big(
 \exp\_{x}^{-1}(y)
 +
 \exp\_{y}^{-1}(x)
 \big)
$$
where $\exp\_z$ is the Riemmannian exponential map at $z\in \{x,y\}$ and $d\_{\mathbb{H}^n}$ is the geodesic/length metric. What is any peer-reviewed (in the mathematics community) reference which shows this/where I can refer to?



Reference: parallel transport in the hyperboloid model
I doubt you will find this in a modern "peer-reviewed" work: deriving the formula is suitable as a homework exercise for a course in semi-Riemannian geometry.


Here's a sketch of proof:


1. The hyperboloid model realizes $\mathbb{H}^n$ as a hypersurface in Minkowski space $\mathbb{R}^{1,n}$. This hypersurface is totally umbilic.
2. Totally umbilic hypersurfaces $\Sigma$ of a semi-Riemannian manifold $M$ have the following nice property: if $\gamma$ is a geodesic in $\Sigma$ through a point $p$, and $v\in T\_p\Sigma$ is such that $v$ is orthogonal to $\gamma$, then the parallel transport of $v$ along $\gamma$ relative to the geometry of $\Sigma$ is the same as the parallel transport of $v$ along $\gamma$ relative to the geometry of $M$.


*Sketch of proof:* since $\gamma$ is geodesic and $v$ is parallel transported, it is always orthogonal to $\gamma$. But as $v$ is orthogonal to $\gamma$, and $\Sigma$ is umbilical, the second fundamental form $II(\dot{\gamma},v) = 0$. Hence parallel transport in $\Sigma$ agrees with the ambient parallel transport.
3. In general, if you parallel transport a vector $v$ along a geodesic $\gamma$, you have that $\frac{d}{dt}\langle v,\dot{\gamma}\rangle = 0$.
4. Ambient parallel transport of $\mathbb{R}^{1,n}$ leaves vectors constant.
5. So in your case: $\exp\_x^{-1}(y) / d(x,y)$ is the unit vector at $T\_x$ that is pointing in the direction of $y$. The above discussion shows that after parallel transporting, the components of $v$ that are orthogonal to $\exp\_x^{-1}(y)$ remains the same, and the component in the direction of $\exp\_x^{-1}(y)$ gets mapped to a vector in the direction of $-\exp\_y^{-1}(x)$ (this being the parallel transport of $\exp\_x^{-1}(y)$ along said geodesic from $T\_x$ to $T\_y$) with the same length. And the result follows.


Incidentally, the same computations work for every hyperquadraic embedded in arbitrary semi-Riemannian vector spaces. So it works for the sphere too.


