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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].

765 questions
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### 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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### (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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### 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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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 $v=... 0answers 133 views ### Extending Green's theorem from very special regions to more general regions Green's theorem Let$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$... 1answer 79 views ### Same fiber of induced covering map [closed] Consider a holomorphic map$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-... 0answers 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 ... 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 ... 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 ... 0answers 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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### 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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### 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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### 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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