# Questions tagged [jet-bundles]

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### What exactly is $T^2f$?

Given a smooth map $f : M \to N$ between manifolds, the differential gives a map $Tf : TM \to TN$ between tangent bundles. Taking another differential gives a map $T^2f : T^2M \to T^2N$ between ...
• 2,406
1 vote
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### What does the "$k$-th tangent mapping" refer to?

I'm reading the section on jet bundles in the book Manifolds of Differentiable Mappings. The author defines a k-jet from $X$ to $Y$ as an equivalence class of pairs $$(f : X \to Y, x \in X),$$ where ...
• 2,406
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### Valid interpretation of Higher order frame bundles and their jet groups?

I've been trying to develop an intuition for higher order frame bundles to help me understand them and this is what I've come up with. Criticisms welcome, as I'm not sure it's valid? NOTE: Always I ...
• 338
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### Condition for section of the frame bundle of $M$ to be the 1-jet of a local diffeomorphism $\phi: M \rightarrow \mathbb{R}^n$?

We're given an n-dimensional Riemannian manifold $M$ and its frame bundle $FM$. The tangent bundle can be locally regarded as an invertible map $\phi:M\rightarrow\mathbb{R}^{n}$. Let's confine $\phi$ ...
• 338
1 vote
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• 1,159
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### Easy computation by Ravi Vakil (jet bundles)

I landed on some short notes by Ravi Vakil from the 90s, the Beginner's Guide to Jet Bundles from the Point of View of Algebraic Geometry. The notes are very clear but on the very first page there is ...
• 43
1 vote
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1 vote
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### Proving a general formula for "tangent vectors" (actually $k$-th partial derivatives) when changing coordinates

I want to prove that the order of contact of two functions $f, g: M \to N$ between manifolds is well defined. Definition. Let $f, g : M^{n} \to N^{m}$ be smooth functions. We say $f$ has order of ...
• 6,751
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### Ring structure on a space of functions between vector spaces?

In this Wikipedia article about jets, in the section about rigorous definitions, for the algebro-geometric definition, they take the vector space $C^\infty_p(\mathbb R^n,\mathbb R^m)$ of germs of ...
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