I have a question about Do Carmo notion of horizontal vector (page 79). So he defines natural metric on $TM$ of manifold $M$. Now he chooses vector $V\in T_{(p,v)}(TM)$ and calls $V$ horizontal vector if it is orthogonal to fiber $\pi^{-1}(p)$ under metric of $TM$ (where $\pi :TM\to M$ is natural projection.
What I am confused is that $V$ and $\pi^{-1}(p)$ do not live in a same space, so metric on $TM$ can not receive as input an element of $\pi^{-1}(p)$. Can someone clarify me how should this be understood?