Tangent bundle

In differential geometry, the tangent bundle of a differentiable manifold ${\displaystyle M}$ is a manifold ${\displaystyle TM}$, which assembles all the tangent vectors in ${\displaystyle M}$. As a set, it is given by the disjoint union of the tangent spaces of M. That is,

where ${\displaystyle T_{x}M}$ denotes the tangent space to ${\displaystyle M}$ at the point ${\displaystyle x}$. So, an element of ${\displaystyle TM}$ can be thought of\as a pair ${\displaystyle (x,v)}$, where ${\displaystyle x}$ is a point in ${\displaystyle M}$ and ${\displaystyle v}$ is a tangent vector to ${\displaystyle M}$ at ${\displaystyle x}$. There is a natural projection

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