Consider an $n$-dimensional complex manifold $M\subset\mathbb{C}^N$ and let


$$f:\mathcal{U}\subset\mathbb{C}^n\rightarrow \mathcal{V}\subset M\subset\mathbb{C}^N$$


be a local parametrization of $M$.


Assume that for all $p\in\mathcal{U}$ we have that $f(p)$ is a linear combination of $\frac{\partial f}{\partial x\_1}(p),\dots, \frac{\partial f}{\partial x\_n}(p)$.


Is then $M\subset\mathbb{C}^N$ an affine subspace of $\mathbb{C}^N$?


In general, does there exist a term for a point $q = f(p)\in M$ such that $f(p)$ is a linear combination of $\frac{\partial f}{\partial x\_1}(p),\dots, \frac{\partial f}{\partial x\_n}(p)$?



Manifolds whose tangent spaces have a special behavior
The answer to the first question is **No**.


The assumption you made is equivalent to stating that for every $q\in M$ that the vector $q\in T\_qM$.


This is satisfied whenever $M$ is a portion of a cone, which need not be affine.


