I have two manifolds $E_{n}=\{([0:x_{1}:x_{2}],[y_{1}: y_{2}])\in \mathbb{C}\mathbb{P}^{2}\times \mathbb{C}\mathbb{P}^{1}\}$ e $V_{n}=\{([x_{0}:0:x_{2}],[y_{1}: y_{2}])\in \mathbb{C}\mathbb{P}^{2}\times \mathbb{C}\mathbb{P}^{1}\}$ and I need to found a base for $T_{[0:0:1],[0:1]} E_{n}$ and for $T_{[0:0:1],[0:1]} V_{n}$, their tangent spaces. How can I do ?
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What is your definition for tangent space? Do you know how find the basis for e.g. $T_{[0,1]} \mathbb{CP}^1$? – Aug 26 '14 at 14:19
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I know that the tangent space is generated by $(\frac{\partial}{\partial x_{i}})$, where $x_{i}$ are the local coordinates of the manifold in p but I don't know how to calculate the basis practically – user166147 Aug 26 '14 at 15:33
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Then can you find a local coordinate for $E_n$ and $V_n$? – Aug 26 '14 at 15:35
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$(0,\frac{x_{1}}{x_{2}},\frac{y_{1}}{y_{2}})$ and $(\frac{x_{0}}{x_{2}},0,0)$? – user166147 Aug 26 '14 at 15:40