# Questions tagged [differentials]

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22 questions
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### Closed forms and trajectories of vector fields

This question is inspired by this recent one and this one; I hope it's not too elementary. Let $M$ be a (closed) smooth manifold and $X$ a vector field on $M$. Fix any Riemannian metric $g$ on $M$ ...
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### Relationship between double tangent bundle, exterior derivative and connection

I am totally new to the subject differential geometry, and that probably reflects itself in the naive question that I'm trying to formulate. I hope this question does not get closed because of this. ...
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### Geodesics for a Cone Metric

Here is a question that I hope/suspect is elementary but cannot find a reference for. Suppose we are given a surface, S, with a conformally Euclidean metric, |f(z)||dz|, where f(z) is meromorphic. ...
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### When does the relative differential $df=0$ imply that $f$ comes from the base?

Let $A \to B$ be a map of commutative rings, and $d : B \to I/I^2$ be defined by $df = f\otimes 1 - 1\otimes f$, where $I$ is the kernel of $B \otimes_A B \to B$, as in [Hartshorne II.8]. If $df=0$,...
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### Compute differential on cotangent bundle

Hi, This is my question. Can we compute easily the differential of the following map ? $$f:(x,\xi^\star)\in TS^{2n-1} \mapsto \xi^\star(ix)\in \mathbb{R}$$ where $TS^{2n-1}$ is ...
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### Various flavours of infinitesimals

I'm not sure if this is a soft question, and whether it may be too broad or, on the contrary, too localized. Well, in Mathematics the concept of "infinitesimal" has been of extreme importance for ...
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### If $\Omega_{X/Y}$ is locally free of rank $\mathrm{dim}\left(X\right)-\mathrm{dim}\left(Y\right)$, is $X\rightarrow Y$ smooth?
Suppose I have a morphism $f:X\rightarrow Y$ such that the relative sheaf of differentials $\Omega_{X/Y}$ is locally free. Does it follow that $f$ is smooth? The answer is no, but for a silly reason. ...