Questions tagged [smooth-manifolds]
For questions about smooth manifolds, a topological manifold with a maximal smooth atlas.
6,835 questions
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How are these two (seemingly) different definition of the differential compatible?
This can be viewed as a follow up question to this question I asked yesterday. In the linked question, given a smooth manifold $M$ and $f\in C^{\infty}(M)$ I asked for clarification on how can the ...
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Why is it good idea to construct the tangent bundle to define vector fields on a manifold rather than as maps to $\mathbb{R^n}$ [closed]
Due to way we have to defined the tangent bundle, from a manifold picture, the plane with usual structure cannot have, for example, three dimensional vectors assigned to every point. More precisely, ...
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Given a smooth manifold $M$ and $f\in C^{\infty}(M)$, how can the differential $df$ be a $1$-form while acting on vector fields?
I would like to apologies in advanced if some of things I am saying don't make $100$% sense, I am simply very confused.
Given a smooth manifold $M$ and $f\in C^{\infty}(M)$ (meaning a smooth function $...
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Why are the components of a differential 1-form functions on a chart
I am reading Lee's Introduction to smooth manifolds. On page $276$ Lee talks about Covector fields introducing the concept of a differential $1$-forms. On the following page Lee defines;
A (local or ...
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Multivalued "charts-valued" function [closed]
Is it possible to have a global parametrisation of a differentiable manifold where the codomain is multivalued (i.e. at this 2d point $(p_0, p_1)$ these charts $\{(U_i, \phi_i)\}$ are applicable)? The ...
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Gluing SDE solutions in local charts together to form a global solution
In Ikeda and Watanabe's textbook p.249-251, we develop an SDE on a smooth $d$-manifold as follows. We take a Stratonovich SDE in local coordinates
$$dX^i = \sigma_0^i(X_t) dt+\sigma_\alpha^i(X_t) \...
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Intuitive/elementary derivation of the Fubini–Study metric on $\mathbb{C}P^{1}$
In the differential geometry course I am taking at the moment I was asked to find the the metric on $\mathbb{C}P^{1}$ defining the topology (i.e to prove explicitly (without using metrization theorems)...
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Intro to Smooth Manifolds by Lee, Problem 6-13c
The problem asks to prove that:
If $F:N\to M$ and $F':N'\to M$ are smooth maps that are transverse
to each other, then $F^{-1}(F'(N'))$ is an embedded submanifold of $N$ of
dimension equal to $\dim N+...
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What is a PDiff or piecewise smooth manifold?
Wikipedia claims that the inclusion functor $\textbf{PL} \to \textbf{PDiff}$ from piecewise linear to piecewise smooth manifolds is an equivalence of categories. There is also an inclusion $\textbf{...
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On the relevance of nonvanishing gradient of absolute time function in Newtonian spacetime
I'm watching at this lecture by F.P. Schuller on Newtonian spacetime $M$. He requires the condition $dt \neq 0$ everywhere claiming that it allows to write $M$ as the disjoint union $$ M = \dot\bigcup\...
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Does every smooth manifold admit a generic Riemannian metric?
Today, I came across the notion of generic Riemannian metrics for the first time. Some googling around informed me of the "definition" of what it means for a Riemmanian metric to be generic (...
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Is the topology assigned to tangent bundle the initial topology from the projection map?
I'm watching at this lecture by F.P. Schuller. At minute 20.45 he defines the topology assigned to the tangent bundle $TM$ as the initial topology from the topology on the base manifold $M$ under the ...
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On the dimension of integral submanifolds defined by a subset of Killing vector fields closed under Lie bracket (commutator)
I'm aware that the set of Killing Vector fields (KVFs) relative to a given metric tensor field $g$ on a smooth manifold $M$, is a finite dimensional $\mathbb R$-sub Lie algebra of the $\mathbb R$-Lie ...
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Sobolev spaces on pseudo-Riemannian manifolds?
While studying Sobolev spaces, I have seen that one can define them on Riemannian manifolds. Can we similarly define Sobolev spaces on pseudo-Riemannian manifolds? If so, are there any textbooks which ...
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Step function and definition of smooth functions between manifolds
Lee defines smooth maps between manifolds like this:
Let $M$, $N$ be smooth manifolds, and let $F:M\to N$ be any map. We say that $F$ is a smooth map if for every $p\in M$, there exist smooth charts $...
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