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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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        1answer
        227 views

        Volume comparison on Grassmannian

        Let $G(r,n-r)$ be the Grassmannian, which can be identified with the space of all rank $r$ projection matrices in $\mathbb{R}^n$. Let $\mu$ be the uniform measure on $G(r,n-r)$. For any $\lambda_1,......
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        votes
        0answers
        84 views

        Volume ratio of balls in Grassmannian with different metric

        Let $G(r,n-r)$ be the Grassmannian, which can be identified with the space of all rank $r$ projection matrices in $\mathbb{R}^n$. Let $\mu$ be the uniform measure on $G(r,n-r)$. For any $\lambda_1,......
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        votes
        0answers
        55 views

        Uniqueness of Fano varieties

        It is a theorem of Kollár–Miyaoka–Mori that there is a finite number of deformation families of smooth, complex Fano $n$-folds for each $n$ (hence also a finite number of diffeomorphism types). My ...
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        79 views

        (Un)orientable of smooth manifolds of $M^{d+1}$ and its boundary $N^d$ [closed]

        Let $M^{d+1}$ be the ($d+1$)-dimensional smooth manifold with a boundary $N^d$ as a $d$-dimensional smooth manifold. Are these statements true: Statements 1-3: If $M^{d+1}$ is orientable, then $N^d$...
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        votes
        1answer
        102 views

        Are normal coordinates the same as Cartesian coordinates in flat space?

        Let $\gamma_v$ be the unique maximal geodesic with initial conditions $\gamma_v(0)=p$ and $\gamma_v'(0)=v$ then the exponential map is defined by $$\exp_p(v)=\gamma_v(1)$$ If we pick any orthonormal ...
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        1answer
        350 views

        Wu formula for manifolds with boundary

        The classical Wu formula claims that if $M$ is a smooth closed $n$-manifold with fundamental class $z\in H_n(M;\mathbb{Z}_2)$, then the total Stiefel-Whitney class $w(M)$ is equal to $Sq(v)$, where $v=...
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        133 views

        Extending Green's theorem from very special regions to more general regions

        Green's theorem Let $C$ be a positively oriented and consists of a finite union of disjoint,piecewise smooth simple closed curve in a plane, and let $D$ be the region bounded by $C$. If $P$ and $Q$ ...
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        1answer
        79 views

        Same fiber of induced covering map [closed]

        Consider a holomorphic map $h: X \to E$ between compact, connected, complex analytic manifolds Let $p: \tilde{E}\to E$ be the universal cover, and denote by $\tilde{h}: \tilde{X}\to\tilde{E}$ the pull-...
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        54 views

        The idealizer of the space of vector fields with vanishing divergence

        The space of smooth vector fields on a manifold is counted as a Lie algebra with the usual Lie bracket structure. Is there a Riemannian manifold of dimension at least $2$ which satisfies either of ...
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        votes
        1answer
        196 views

        Milnor immersion of circle, disks, and a ball

        Milnor surprisingly found an immersion of a circle in the plane which bounds two "incompatible" immersed disks (see [1] for example, picture included). I think I can glue these two disks together to ...
        4
        votes
        0answers
        121 views

        Integration over a Stiefel manifold

        Let $V_{m}(\mathbb{R}^n)$ be the Stiefel manifold, which is the space of all $n \times m$ matrices such that the $m$ columns are orthonormal. I want to calculate the volume of the following subset of ...
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        30 views

        Integrating over some domain of the Stiefel manifold to analyze its support

        Define the square $d$ dimensional Stiefel manifold as $$V_{d} = \{ R \in \mathbb{R}^{d \times d} : R ^\top R = I_d \} .$$ How does one integrate on this manifold over a domain defined as $\{ R \in V_{...
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        votes
        2answers
        220 views

        Existence of an isotopy in Riemannian manifold

        Let $(M,g)$ be a Riemannian manifold, and $p,q\in M$ be two fixed points. We assume $p,q$ are close enough. Say, we assume $p$ and $q$ are in the same normal coordinate chart. It is clear that there ...
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        0answers
        130 views

        Finite good covers on smooth manifolds

        Let $M$ be a connected smooth manifold that is not necessarily compact but has the homotopy type of a finite CW complex. Does $M$ admit a finite good cover? (i.e. a finite cover by contractible ...
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        64 views

        Infinitesimal matrix rotation towards orthogonality

        TLDR; I am trying to prove the existence of an infinitesimal rotation which always moves a matrix "closer" to being orthogonal. Setting In this setting, we have a matrix $W \in \mathbb{R}^{n \times ...

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