# 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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### Levi-Civita connection from idempotents

Let $(M,g)$ be a closed Riemannian manifold. Let $V$ be a smooth complex vector bundle over $M$. We can write $V$ as the range of an idempotent $E$ in a matrix algebra $M_n(C^\infty(M))$ acting on a ...
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### Cobordism theory of some weird space

Let $G=SU(3)$ and $N=SO(3)$, then $G/N= SU(3)/SO(3)$ = a 5-dimensional Wu manifold $W$. The $W$ is a homogeneous space (also a quotient space), but not a group. Previously, I am aware of the ...
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### $\mathbb Z$-graded manifold is isomorphic to the structure sheaf of supermanifold locally

In this paper, definition 4.4.1 about supermanifold and definition 4.6.1 about graded manifold: Definition 4.4.1: An supermanifold $\mathcal{M}$ is a locally ringed space $(M,\mathcal O_M)$ which ...
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### Every homotopy class contains at least a harmonic representative

Let $(M^3,g)$ be a closed, connected and oriented Riemannian $3$-manifold. A circle-valued map $v : M \to S^1$ is harmonic iff the gradient $1$-form $\omega_v = v^* d\theta \in \Omega_1(M)$ is ...
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### Extend fibre bundle

Let $F\rightarrow E\rightarrow B$ be a smooth fibre bundle. Suppose $W$ is a smooth manifold such that $F=\partial W$. When is it possible to extend the bundle to a bundle over $B$ with fibre $W$?
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### Signature of a non-compact manifold

Let $v_0,\dots,v_n\in\mathbb{Z}^2$ be integer vectors which satisfy the condition $\det\begin{pmatrix}v_{k-1}&v_k\end{pmatrix}=(-1)^k$, whose relevance will become apparent in a moment. We may ...
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### Extend (Lie) group action from the boundary to the entire manifold

Let $M$ and $W$ be smooth manifolds such that $\partial W=M$. Let $G$ be a group acting on $M$. Can one generally extend the action of $G$ to $W$? If not, under which conditions on $W$ and/or $G$ ...
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### Space of representations of surface group into Lie groups

In the context of Goldman's paper The symplectic nature of fundamental groups of surfaces: Consider a closed oriented surface $S$ with fundamental group $\pi$, and let $G$ be a connected Lie group. ...
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### Continuity of $r\mapsto\int_{\Sigma\cap B_r(x)}f^2d\mu$

Let $\Sigma$ be an embedded smooth surface in $\mathbb{R}^3$, and let $f:\Sigma\to\mathbb{R}$ be a smooth function. Suppose $f$ is square-integrable on $\Sigma$, with \begin{align} 0<\int_{\Sigma}f^...
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### Any cobordism invariant made of “characteristic classes”, on unorientable manifolds, must be a mod 2 class?

For any cobordism invariant (or simply bordism invariant) quantity $\omega$ that satisfy the conditions: $\omega$ can be fully decomposed from the cup product of characteristic classes (such as ...
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### Finite-dimensional argument for Morse-Smale pairs?

Using the Sard-Smale theorem, it is relatively easy to show that Morse-Smale pairs on a manifold $M$ (i.e. pairs $(f,g)$ where $g$ is a metric on $M$, $f$ is a Morse function on $M$, and the stable/...
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### Minimizing area in relative homology class

A well known result in geometric measure theory asserts that if $(M^{n+1}, g)$ is a closed Riemannian manifold and $\alpha \in H_n(M)$ is a nonzero homology class, then there exists a closed embedded ...
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### If $M$ is a manifold, $x∈M$ and $d(x,ω)=\inf\{t>0:x+tω∈M\}$, does the pushforward of the solid angle measure under $S^2∋ω↦x+d(x,ω)ω$ admit a density?

Let $S^2$ denote the unit 2-sphere, $M$ be a 2-dimensional oriented embedded $C^1$-submanifold of $\mathbb R^3$ with d_M(x,\omega):=\inf\left\{t>0:x+t\omega\in M\right\}<\infty\;\;\;\text{for ...
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### Confused about A. Kosinski's description about surgery in his book “differential manifolds”

Please excuse me, if MO is not the proper place for this question. I aksed the same question on M.SE https://math.stackexchange.com/questions/3511134/confused-about-a-kosinskis-description-of-surgery-...
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### Why is the divergence theorem used in the Eells-Sampson paper slightly different from that in a textbook?

I am reading Harmonic Mappings of Riemannian Manifolds by Eells and Sampson. In chapter 2, the author(s) used the divergence theorem, which does not look like the usual divergence theorem for ...
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### Relate the solid angle and surface measure of a surface

Let $M$ be a 2-dimensional embedded $C^1$-submanifold of $\mathbb R^3$ with a global chart$^1$ $(U,\phi)$. If $u\in U$ and $x=\phi^{-1}(u)$, let $\nu_M(x)$ denote the unique unit normal vector of $M$ ...