Timeline for Holomorphic retraction $\implies$ holomorphic tubular neighbourhood?
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| Oct 8, 2021 at 12:13 | comment | added | Bad English | In order to make my answer not completely useless let me mention a counter-example to your original question, where the projection definitely gives an isomoprhism. Take $S=\mathbb{P}^1$. Then tangent vectors to $F_x$ ($\tau_r$) are determined by some $r\in Hom(\nu_{S/M},T_S)=\mathbb{C}$. Considering a diagonal action of $SL(2)$ on $M$ we obtain an action on $\mathbb{C}$ which is trivial, hence $\tau_r$ is invariant under the action and at least one of two projections is isomorphism. | |
| Oct 8, 2021 at 12:09 | comment | added | Bad English | Let $\phi:Tot(\nu_{S/M})\to Tube(S/M)$ be any tubular neighborhood. Consider a "retraction" fibers $F_x=\phi^{-1}(x)$ for $x\in S$. Tangent vectors along $F_x$ define a vector bundle $\tau_r\subset {T_M}|_{S}=\nu_{S/M}\oplus T_S$, which is transversal to $T_S$. In other words $\tau_r$ is a deformation of $\nu_{S/M}$ defined by an element in $r\in Hom(\nu_{S/M},T_S)$. If one can produce an automorphism $\alpha$ of $Tot(\nu_{S/M})$ which differential maps $\tau_r$ to $\nu_{S/M}$, then we are done. I don't see how one can do this. | |
| Oct 8, 2021 at 12:08 | comment | added | Bad English | @SimonParker, oops, you absolutely right. I unintentionally took the retraction into account and impose stronger conditions on the definition of a tubular neighborhood. Namely, if the tubular neighborhood morphism restricted to $S$ maps tangent vectors along the normal bundle to tangent vectors along the retraction fibers, then the projection you mentioned above gives an isomorphism and we can obtain a connection. | |
| Oct 6, 2021 at 17:26 | comment | added | Simon Parker | Thanks for your answer. I do not understand the first step of the definition of the connection, namely: "The tubular neighborhood for each $x\in S$ defines identification $f_x$ of a neighborhood $0\in T_x$ with a neighborhood of $x\in S$". The tubular neighbourhood gives an embedding of a neighborhood of $0$ in $T_xS$ in $S \times S$, so presumably you want to compose this with one of the projections to get the desired identification. But it is easy to find examples where this composition is not an isomorphism. | |
| Oct 5, 2021 at 17:48 | history | answered | Bad English | CC BY-SA 4.0 |