Let $X$ be a complex manifold and $Y\subseteq X$ a submanifold. I define $TX_{|Y}$ as the restriction bundle of $TX$. How I can prove that $$d \iota: TY\longrightarrow TX_{|Y}$$ ($\iota:Y\longrightarrow X$ is the canonical inclusion) is injective?
Normal bundle and differential of the inclusion map
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$\begingroup$
algebraic-geometry
vector-bundles
tangent-bundle
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1Using what definition of tangent space? Do you understand the geometric picture for why this should be true? Maybe first show that the induced map on tangent vectors is injective (as a map between tangent spaces - i.e. the differential of the inclusion). Then relate it to the pullback bundle. – 2017-01-20
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0I use the canonical def with tangent vectors. Can you expand the answer? – 2017-01-21