# 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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### $h$-convexity requirement: does taking the directional derivative over a curve preserve inequality?

Context: The result below implies that a certain class of random variables form a non-negative supermartingale bounded by 1, with minimal assumptions on the distribution of the random variables, which ...
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### Derivative norm estimates

Assume $\Phi$ is some diffeomorphism of a certain manifold. Let $\Phi^{-1}$ denote the inverse map and let $(D\Phi)^{-1}$ denote the matrix inverse of $D\Phi$. QUESTION. Does this norm estimate hold? ...
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### Dual space of local Sobolev space on a manifold

$\newcommand{\comp}{\mathrm{comp}}$As part of my master's thesis, I am currently learning about Sobolev spaces on manifolds. From my research online, I found out, that there are a lot of ways to ...
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### Property of a smooth function is true for almost every function

Consider a smooth function $f:\mathbb{R}^n\to \mathbb{R}$. We further assume $0$ is a regular value of $f$, thus $X=f^{-1}(0)$ is a smooth submanifold of dimension $n-1$. Consider the following ...
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### Identify an SDE on the sphere from its generator

I have a diffusion on the 2-sphere with expression: $$(L\phi)(u):=\frac{1}{2{N(u)}}\Big(f(u)\Delta_{\mathbb S^2}\phi+ 2g\left( \nabla_{\mathbb S^2}\phi, \nabla_{\mathbb S^2}f\right)\Big)$$ ...
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### No analytic surjection $f:M \to N$ when $\dim(M) >\dim(N)$

Inspired by comment discussions in this MO post smooth version of splitting principle we ask: Are there two compact real analytic manifolds $M,N$ of dimension $m>n$ such that there is not any ...
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1 vote
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### Smooth version of the splitting principle

Inspried by this MO question A manifold whose tangent space is a sum of line bundles and higher rank vector bundles we pose the following question as a possible smooth version of the splitting ...
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### When can a generalized connected sum be aspherical

Let $M$ and $N$ be compact $n$-dimensional manifolds with a common (nicely embedded) compact submanifold $S$ (we may assume that the inclusion of $S$ in $M$ and $N$ is $\pi_1$-injective). Let $M\#_S N$...
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### Unimodular intersection form of a smooth compact oriented 4-manifold with boundary

Let $X$ be a smooth compact oriented 4-manifold with nonempty boundary. Its intersection form $$Q_X : H^2(X,\partial X;\Bbb Z)/\text{torsion}\times H^2(X,\partial X;\Bbb Z)/\text{torsion}\to \Bbb Z$$ ...
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### Topological Properties of Subsets of $R^{m}$ induced by Smooth Manifolds in Matrix Spaces

We know that $M_{m \times n }$ is isomorphic to $R^{mn}$. Let's take a smooth manifold $\mathbb{M}$ in $R^{mn}$ and fix a point in $R^{n}$, say, $p$. Now realize the manifold $\mathbb{M}$ as a subset ...
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### Extending diffeomorphisms between surfaces

Suppose we have two smooth compact oriented surfaces $M_1$ and $M_2$ with boundary,both of them have genus $g$, and there are orientation preserving diffeomorphisms $\psi_1, \psi_2, \cdots, \psi_n$, ...
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1 vote
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### Can you recover the manifold diffeology from the diffeology of distributions?

Is it true that the manifold diffeology is the subspace diffeology heredited from the diffeology of distributions? I wanna know the same thing for the tangent bundle.
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### Where is the Steenrod Realization problem at?

I'm wondering if there is a more modern reference out there for the Steenrod Realization Problem than the book of Connor and Floyd? Realizing homology classes in a manifold via embedded submanifolds, ...
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### How is the $k$-times iterative frame bundle $FF\cdots FM$ associated to the higher order frame bundle $F^k M$?

$\DeclareMathOperator\Gl{Gl}$As I understand it a natural bundle is one for which a diffeomorphism on the base space lifts to an automorphism on the total space of the bundle. It is my understanding ...
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### Estimate the gradient (with respect to local coordinates) of a partition of unity on a manifold

Suppose $\{U_\alpha\}$ is an atlas of coordinate patches of a (noncompact) smooth manifold $M$ of dimension $n$, with coordinates $(x_\alpha^1,\dots,x_\alpha^n)$ on $U_\alpha$. Furthermore we assume ...
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### Noether's formula for real algebraic surfaces

Is there a version of Noether's formula for the Euler characteristic of a surface for Real algebraic surfaces? Specifically, given $X$ a real algebraic compact smooth surface, what is the relationship ...
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### Does a coarser topology lead to a non-Hausdorff topological manifold? [closed]

Take a topological manifold $M$. Suppose one considers a strictly coarser topology than the manifold topology. Can such topology result in a non-Hausdorff topological manifold? NOTE: PLEASE avoid the ...
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### What is known about the "stickiness" of a smooth manifold to its tangent space?

Consider an $m$-dimensional smooth manifold $M$ embedded in $n$-dimensional real Euclidean space $\mathbb{R}^n$ ($n>m$). Consider a point $x \in M$ of the manifold, and for simplicity, choose the ...
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### Assumptions for uniform measure of SDE on manifolds

Suppose we're working on a compact, Riemannian manifold $M$. Suppose $dX_t = -b(X_t, t)\,dt + \sigma^2 \,dB_t$ is started at the uniform measure on $M$. What kind of assumptions on $b$ make it so that ...
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### Smooth compactification of complex varieties and uniqueness

Since I'm working in differential geometry, for the following I'm strictly interested in the smooth setting over $\mathbb{C}$ and its relation to the setting over $\mathbb{R}$. Here are a few useful ...
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Let $X$ and $Y$ be smooth oriented manifolds of dimension $m$ and $n$, and let $f\colon X\to Y$ a proper smooth map. There is a theorem called the "Disintegration Theorem" which says ...