Questions tagged [manifolds]
For questions on manifolds of dimension $n$, a topological space that near each point resembles $n$-dimensional Euclidean space.
9,148 questions
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Borel-$\sigma$ algebra on a manifold
Let M be an n-dimensional manifold, and let $(\pi_\alpha: U_\alpha \to V_\alpha)$ be an atlas of coordinate charts for M, where $U_\alpha$ is an open cover of M and $V_\alpha$ are open subsets of ${\...
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Special case of Whitney Extension Theorem: Extension of $C^r$ function on a relative open set to a open set.
A special case of Whitney Extension Theorem says:
Let $U$ be a relative open set in $\mathbb{H}^k$. Let $V$ be the (absolute) interior of $U$ and let $f\in (U, \mathbb{R})$. Then the following ...
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Cohomology classes on manifolds induce homology and bordism classes of Eilenberg-MacLane spaces
Let $M$ be a closed oriented smooth manifold of dimension $d$, and let $G$ be an abelian group. Then the cohomology group $H^n(M;G)$ corresponds bijective to set of homotopy classes of based maps $[M,...
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The limit of a family of vector fields is independent of chart
This is problem 20.1 from Loring Tu's An Introduction to Manifolds:
Let $I$ be an open interval, $M$ a manifold, and {$X_t$} a 1-parameter family of vector fields on $M$ defined for all $t \neq t_0 \...
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Does online mirror descent between dually flat space converge to the global optimum
Say $E = \{p_\theta : p_\theta(x) = \exp(x^\top \theta - A(\theta)), \theta \in \Theta, M \theta = b\}$ is an exponential family affinely constrained in its natural parameter, where $\Theta$ is a ...
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The directional derivative in local coordinates - definition verification
I am currently studying Lie groups and manifolds using these notes. As a follow up to this question (in which I asked how the partial derivative of a regular function is defined with respect to local ...
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Understanding the application of FTC in the proof that $\dim \left( T_{p}M \right) = n$
Consider the following proof I am trying to break down
There are two things I don't understand in the proof.
I do not understand the way the author uses the Fundamental Theorem of Calculus. I know ...
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Understanding the definition of paracompactness
Recently I came across the definition of paracompactness (while reading about manifolds).
Here are the relevant definitions for paracompactness:
Let $X$ be a topological space.
Definition (cover, open ...
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Does a dense orbit imply topological transitivity for flows on manifolds?
Let $M$ be a smooth connected manifold of dimension $\geq 2$, and let $\phi: \mathbb{R} \times M \to M$ be a complete flow on $M$.
Suppose there exists a point $x_0 \in M$ whose orbit is dense in $M$, ...
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Regarding proof that Busemann functions are precisely all the horofunctions in a Hadamard manifold
I am following Manifold of nonpositive curvature by Ballmann, Gromov and Schroeder. I am trying to understand the proof of Lemma 3.4 in page 26.
Lemma 3.4. For $h \in C(X)$. The following are ...
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Proving the set of zeros of a $F: \mathbb{R}^{n} \to \mathbb{R}^{m}$, $C^{1}$ func is a $n-m$ dimensional manifold with the Implicit function theorem
This is a follow up question to this question I asked earlier today, in which I asked whether the function $g$ we get from the implicit function theorem is a homeomorphism. As pointed out in the ...
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Doubt about the formulation of the implicit function theorem (and its application to manifolds)
Notation
Let $O \subseteq \mathbb{R}^{n+m}$ be open and denote $\mathbf{x} = \left( x_1,\ldots,x_{n}\right) \in \mathbb{R}^{n} , \mathbf{y} = \left( y_1,\ldots,y_{n}\right) \in \mathbb{R}^{m}$. In ...
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Can you put a (distorted) grid on a compact manifold?
Let $M$ be a compact, $d$-dimensional topological manifold. Let $I^d$ denote the $d$-cube and let $I^d_n$ denote the $(2^n+1)^d$ dyadic rationals of precision $n$ contained in $I^d$ (i.e. the points ...
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Possible Error in O'Neill's Semi-Riemannian Geometry
I'd like to confirm what I think may be an error in O'Neill's Semi-Riemannian Geometry to ensure I'm not missing something. Chapter 7, exercise 11 claims that if $B \times_f F$ is a warped product, a ...
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Proving that $T_xU$ and $T_xM$ are isomorphic tangent spaces
I'm following John M Lee's "Introduction to smooth manifolds" and came across the following.
"Let $M$ be a smooth manifold and $U\subset M$ an open subset of $M$. Then there is a vector ...
