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        Questions tagged [smooth-manifolds]

        Smooth manifolds and smooth functions between them. For manifolds with additional structure, see more specific tags, such as [riemannian-geometry]. For more topological aspects, see [differential-topology].

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        Matrix trace minimization of quadratic and linear terms under orthogonal manifold constraints

        How would one solve the following orthogonal manifold problem? $\max_{\{X : X^\top X = I\}} \text{tr}(X^\top A X - X^\top B)$ where $A \succeq 0$ I've seen one method that successively performs the ...
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        Structural Stability on Compact $2$-Manifolds with Boundary

        I'm studying the structural stability of vector fields and I'm interested in learning about this phenomenon on compact $2$-manifolds with boundary. Let $M^2$ be a compact connected 2-manifold and $\...
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        1answer
        285 views

        Smooth vector fields on a surface modulo diffeomorphisms

        Let $\Sigma$ be a two-dimensional connected smooth manifold without boundary. (Feel free to assume it is compact and orientable.) Let $\mathcal{X}(\Sigma)$ denote the smooth vector fields on $\Sigma$...
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        237 views

        Integration over a Surface without using Partition of Unity

        Suppose we are given a compact Riemann surface $M$, an open cover $\mathscr{U}=\{U_1,U_2,\dots\}$ of $M$, charts $\{(U_1,\phi_1),(U_2,\phi_2),\dots\}$, holomorphic coordinates, $\phi_m:p\in U_m\mapsto ...
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        203 views

        On the homotopy type of $\mathrm{Diff}(\mathbb{S}^3)$

        I am confused with the following argument. I know I am doing something wrong but I can't find my mistake. On one hand, one knows that if $M$ is a Lie group, then $$\mathrm{Diff}(M)\simeq M\times\...
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        Traceless sobolev forms on compact manifolds with boundary

        Let $(M,g)$ be a smooth, compact, oriented Riemannian manifold with smooth, oriented boundary $\partial M$. Further, let $\Omega^p(M)$ and $\Omega^p(\partial M)$ be the spaces of smooth differential $...
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        1answer
        327 views

        Metrics on derived smooth manifolds

        Derived geometry explains how to remove the transversality condition and make sense out of a nontransversal intersection. For example, if $X$ and $Y$ are embedded submanifolds of a manifold (or ...
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        History of the definition of smooth manifold with boundary

        I am trying to determine the earliest source for the definition of smooth ($C^\infty$) manifold with boundary. Milnor and Stasheff (1958) give a definition, but a scrutiny of that definition shows it ...
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        71 views

        Quartic link in a 5-sphere

        In this post I would like to propose a quartic link in a 5-sphere. Let us start with the following gluing into a 5-sphere: $$S^5=(D^2_{} \times T^3_{}) \cup_{T^4} ({S^5 \smallsetminus D^2 \times T^3})...
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        161 views

        Triple link in a 5-sphere — Proposal

        In this post I would like to propose a triple link in a 5-sphere. Let us start with the following gluing into a 5-sphere: $$S^5=(D^2_{} \times T^3_{}) \cup_{T^4} ({S^5 \smallsetminus D^2 \times T^3})$...
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        1answer
        171 views

        Are complete minimal submanifolds closed?

        Is it true that any complete minimal submanifold of some Riemannian manifold is closed as a subset? What about the case in which the ambient manifold is an euclidean space?
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        Condition on a Lie groupoid to be represented by manifold/group or an action groupoid

        Let $\mathcal{G}$ be a Lie groupoid. I am thinking of following questions. When do we know $\mathcal{G}$ is weakly/Morita equivalent to a Lie groupoid of the form $(G\rightrightarrows *)$ for some ...
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        1answer
        308 views

        Are framed manifolds cubulatable?

        Let's say an $n$-manifold is cubulated if it is glued out of cubes $[0,1]^n$ in a way that looks locally like the standard cubulation of $\mathbb R^n$. For instance, the face $[0,1]^{k-1} \times \{1\} ...
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        1answer
        92 views

        Smooth embedding of space forms in the Euclidean space

        I was wondering which $S^n/\Gamma$ can be smoothly embedded into $\mathbb R^{n+1}$, where $\Gamma \subset O(n+1)$ is a finite subgroup. To my knowledge, the case $n \le 3$ is known. It has been proved ...
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        139 views

        $k$ times differentiable but not $C^k$ manifold

        I asked the following question on Math Stack Exchange 3 months ago but got no answer. So maybe Math Overflow is a more suitable place for such a question: I cannot find the notion of $k$ times ...

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