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The tangent bundle $TM$ of a manifold $M$ is defined as the disjoint union of all tangent spaces $T_pM$ for every point $p$ in $M$. Symbolically, $TM = \bigsqcup_{p \in M} T_pM$.

Conditions: $M$ is a smooth manifold.; $T_pM$ is the tangent space at point $p$.

The tangent bundle $TM$ of a manifold $M$ is defined as the disjoint union of all tangent spaces $T_pM$ for every point $p$ in $M$. Symbolically, $TM = \bigsqcup_{p \in M} T_pM$.

Conditions: $M$ is a smooth manifold.; $T_pM$ is the tangent space at point $p$.

A covariant 2-tensor field is linear over the ring of smooth functions. If $X, Y$ are vector fields and $f, h$ are smooth functions, then $g(fX, hY) = f h g(X,Y)$.

Conditions: $g$ is a covariant 2-tensor field.; $X, Y$ are vector fields.; $f, h$ are smooth functions.

A covariant 2-tensor field is linear over the ring of smooth functions. If $X, Y$ are vector fields and $f, h$ are smooth functions, then $g(fX, hY) = f h g(X,Y)$.

Conditions: $g$ is a covariant 2-tensor field.; $X, Y$ are vector fields.; $f, h$ are smooth functions.