The curve-based <tangent space> at $x$ consists of velocities $\gamma'(0)$ of smooth curves $\gamma:(-\varepsilon,\varepsilon)\to M$ satisfying $\gamma(0)=x$. For an arbitrary subset these velocities need not form a <vector space>; for a smooth <submanifold> they do, and this is its usual <tangent space>. This definition suffices for the direct computation at the identity.

Solution

Past exam of the short options course of the University of Oxford / 2023 / 4 / b / i / Solution

 Solution (source code)