# 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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### Does composition on the right by a volume-preserving diffeomorphism preserve homotopy class?

Let $M, N$ be smooth manifolds with $M$ orientable and compact. Let $\sigma$ be some volume form on $M$ and consider the set $\mathcal{M}$ of smooth maps from $M$ to $N$ in a fixed homotopy class. Now ...
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### S¹ action on a manifold which generates "non-torsion" loop in diffeomorphism group

I am interested in $S^1$-actions on smooth, closed, and oriented manifolds $M$. I suppose that the action has a fixed point (I also suppose $M$ is connected). Let $\operatorname{Diff}(M)$ denote the ...
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### First usage of the terms pseudo-isotopy and concordance in manifold theory

I am hoping I can use the collective knowledge of the forum to piece together some history. I'm wondering where the terms pseudo-isotopy and concordance originated, in their modern forms as used in ...
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### Proving an equality of differential forms by assuming some perhaps topological condition

Let say I want to show two differential forms $\omega_1$ and $\omega_2$ on a smooth manifold $M$ are equal. Of course it suffices to show $\omega_1=\omega_2$ locally, i.e. the equality holds over ...
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### Diagonalization of symmetric matrices of functions

I asked this question some time ago in MSE but I didn't recieved any feedback. https://math.stackexchange.com/questions/4672664/diagonalization-of-symmetric-matrices-of-functions This problem arised ...
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### Is the intersection of such a triple of minimal surfaces in the 3-ball a single point?

Let $S_1,S_2,S_3$ be three simple closed curves on the 2-sphere $\mathbb{S}^2$. (With no smoothness or rectifiability assumption) For each $i$, let $M_i$ denote the minimal surface (i.e. disc) bounded ...
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### Difference between parallel transport and ambient projection

Consider a $d$-dimensional complete embedded Riemannian submanifold $(M,g)$ of a Euclidean space $\mathbb{R}^D$ (The major examples we consider are sphere and Stiefel manifold). Assume the sectional ...
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### Integral flow that can commute to Laplacian operator

Firstly, considering the vector field in $\mathbb{R}^3$, $X=x_2e_1-x_1e_2$, we can see that $$\phi(t,x)=\phi(t,x_1,x_2,x_3)=(t,x_1\cos t+x_2\sin t,-x_1\sin t+x_2\cos t,x_3)$$ is the ...
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### Applications of Strong Whitney Embedding

I am looking for applications of the strong Whitney's embedding theorem that have an advantage over weak theorems. That is, applications where it's important that the dimension of the Euclidian space ...
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### Are these two concepts of a differential form on the loop space equivalent?

Notation: Let $X$ denote a smooth manifold (without boundary) and define $LX = C^{\infty}(S^1, X)$ to be the loop space on $X$. In the context of loop space homology and the supersymmetric path ...
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### Leaves of bounded genus

Let $\mathcal{F}$ be a codimension one foliation in a closed $3$-manifold $M$. Does there exist an upper bound for the genus of the compact orientable leaves? That is, does there exist $G >0$ such ...