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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?

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If $A$ is an affine space, and $x \in A$, then there is a natural identification of $A$ with $T_xA$. The space $\pi^{-1}(p)$ is a submanifold of $TM$. Thus the tangent space at $(p,v) \in TM$ is the direct sum of the tangent space $T_{(p,v)}(\pi^{-1}(p))$ and a 'horizontal' vector space. The former we can identify with $\pi^{-1}(p)$ itself, according to the first sentence of this comment. – yasmar Oct 10 '12 at 5:21

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