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        Questions tagged [manifolds]

        A manifold is a topological space that locally resembles Euclidean space near each point. More precisely, each point of an n-dimensional manifold has a neighbourhood that is homeomorphic to the Euclidean space of dimension n.

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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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        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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        220 views

        Can a hyperbolic manifold be a product?

        I was interested in whether a manifold which admits a metric of constant sectional curvature can be homotopy equivalent to a product of non-contractible manifolds. Of course, there are three cases: ...
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        255 views

        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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        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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        150 views

        harmonic coordinates on non-compact manifolds

        Is it possible to show the existence of harmonic coordinates (e.g., on uniform-sized balls) on certain classes of non-compact Riemannian manifolds? For example, one may expect that such harmonic ...
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        2answers
        109 views

        Self-adjoint extensions for pseudo-differential operators

        The class $\Sigma^1$ of symbols on $\mathbb R^{2n}$ is made with $C^\infty$ functions $a$ of $X=(x,\xi)\in \mathbb R^n\times\mathbb R^n$ such that $$ \vert\partial_X^\alpha a\vert\le C_\alpha(1+\vert ...
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        58 views

        Vector bundle endomorphism diffeomorphism invariant?

        Let's say we have a vector bundle $V$ on a compact Riemannian manifold $M$ (of dimension $m$, with metric $g$). Given a differential operator $P$, acting on the sections of $V$, of the form: $$P = \...
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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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        79 views

        Manifold with no closed components?

        Let $M$ be a manifold with boundary. Reading some papers on $3$-manifolds I have come across some statements where they require that: ”$M$ has no closed components.” What does this mean? The ...
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        216 views

        Is it possible to glue together complex manifolds?

        In the case of Riemannian manifolds, there are ways to take two manifolds and glue them together to get a new Riemannian manifold. For example, taking connected sums in local regions where the two ...
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        1answer
        723 views

        Can one determine the dimension of a manifold given its 1-skeleton?

        This may be an easy question, but I can't think of the answer at hand. Suppose that I have a triangulated $n$-manifold $M$ (satisfying any set of conditions that you feel like). Suppose that I give ...
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        28 views

        dual and intersection of a simplex

        In a triangulation $\Gamma$ of a (oriented) 2-manifold, consider a 2-simplex labeled by ($123$), where $1,2,3$ denote the order of vertices. Consider the dual $\Gamma^*$ of $\Gamma$, and then denote ...
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        0answers
        153 views

        Reference for a proof of a Theorem by Joseph Wolf

        We know that Lie Groups are parallelizable, I was looking for a version of the converse and came across this: https://books.google.com/books?id=w4bhBwAAQBAJ&pg=PA115 in Introduction to Smooth ...
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        1answer
        385 views

        Critical dimensions D for “smooth manifolds iff triangulable manifolds”

        I am aware that at least for lower dimensions, "smooth manifolds iff triangulable manifolds" at least for dimensions below a certain critical dimensions D. My question is that for ...

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