Example 17.10.23C16140E8B047

Example 17.10.23.C16140E8B047 Let $\varphi _0\in \mathrm{PSH}(X,\theta )$ and $C\in \mathbb {R}$. Then the subgeodesic $(\varphi _0+tC)_{t\in [0,1]}$ studied in example 17.10.6 is a geodesic. This follows easily from proposition 17.10.7.

In particular, when $C=0$, we find that the constant subgeodesic at $\varphi _0$ is indeed a geodesic, which we call the constant geodesic at $\varphi _0$.

More generally, suppose that $(\varphi _ t)_{t\in [0,1]}$ is a geodesic and $C_1,C_2\in \mathbb {R}$, then $(\varphi _ t+C_1t+C_2)_{t\in [0,1]}$ is also a geodesic. This follows immediately from example 17.10.6.