What is an example of a $n$ dimensional manifold $M$ which is not a lie group or $S^{7}$ but satisfies the following property?:



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> There is an $n$ dimensional sub vector space $V\subset \chi^{\infty}(M)$ such that every $0 \neq X \in V$ is a nonvanishing vector field on $M$.
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So this is a motivation to define an invariant of manifolds as follows:
$$ \text{The maximum number $k\leq n$ with a k dimensional subvector space$$ $V\subset\chi^{\infty}(M)$ with the above property} $$



A full dim. subvector space of $\chi^{\infty}(M)$ which all non zero elements are nonvanishing vec.field
These manifolds, which have a trivial tangent bundle, are called parallelizable. They include all open sets of $\bf R^n$, thus I assume that you had in mind closed (compact without boundary) manifolds.


There are still many closed parallelizable manifolds, the most famous being perhaps all three-dimensional orientable manifolds (Stiefel's theorem, see Milnor-Stasheff, *Characteristic classes*, Problem 12-B page 148). The simplest example is the product $S^1\times S^2$.


