# Tagged Questions

For questions on manifolds of dimension $n$, a topological space that near each point resembles $n$-dimensional Euclidean space.

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### Exterior derivative for functions with values in a parallelizable manifold

In Sharpe's text on Cartan geometry, he explains in section 1.5 on page 52 how to define an exterior derivative for maps into a parallelizable manifold $N$. Let $f: M \to N$ be a smooth map, and ...
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### Dimension of topology manifold

In the 3 page of Jurgen Jost's Riemannian Geometry and Geometric Analysis .Why it is harder in topology manifold than differentiable manifold ? I think it is easy in differentiable manifold because ...
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### Intersection of kernels of linearly independent smooth 1-forms on $\mathbb R^n$

I'm trying to solve the following problem: Let $\omega^1,\dots,\omega^k$ be smooth $1$-forms on $\mathbb R^n$ that are linearly independent at each point of $\mathbb R^n$. For $p\in\mathbb R^n$, ...
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### An orientation on a $(n-1)$-dimensional submanifold.

This question goes on where this question ended. Given is an non-empty $(n-1)$-dimensional ($n\ge2$) differentiable submanifold $X\subset\mathbb{R}^n$ such that there exists an open ...
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### Orientability of the level set of a map between abstract oriented manifold

Let M and N be oriented manifold and let $f:M\to N$ be a smooth map between them. Suppose $y \in N$ is a regular value for $f$, how can we show that $f^{-1}(y)$ is orientable? I've seen a solution ...
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### Topological boundary of a specific manifold.

Let $\emptyset\neq X\subset\mathbb{R}^n$, $n\ge2$, be an $(n-1)$-dimensional differentiable submanifold, i.e. for every $p\in X$ there is an open $U_p\subset\mathbb{R}^n$ with $p\in U_p$, and a ...
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### connected sum of surfaces is well defined proof attempt

Suppose $S_1$ and $S_2$ are compact surfaces (connected 2-dimensional manifolds). If we cut out of them two closed disks, and glue the surfaces along disk boundaries we get new surface, their ...
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### Proof Writing: Given 2 two dimensional manifolds $\emptyset \neq N \subseteq M \subseteq \mathbb{R}^3$, $N$ is compact, $M$ is pconnected: $N = M$

Statement: Given 2 two-dimensional manifolds $\emptyset \neq N \subseteq M \subseteq \mathbb{R}^3$, if $N$ is compact and $M$ is path-connected, then $N = M$. Proof: We know that there is at least ...
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### Brownian motion on a manifold

If I have a manifold $M$ and a chart $\left(x,U\right)$, is it possible to simulate Brownian motion on that manifold by solving an SDE in the chart representation $x\left(U\right)$ and then use the ...
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### Why are tangent vectors coordinate-dependent?

Why does the coordinate basis for $T_pM$ depend on the coordinate chart we are using? Any two charts containing $p$ agree on some neighborhood of $p$, so shouldn't we be able to find a basis for ...
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### Why are PDEs with Hamiltonians usually solved on compact manifolds?

The title is self explaining: I see in a lot of literature that PDEs with some Hamiltonian structure in it are solved over a torus or some other compact manifold. Why is that? At least I now that it ...
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### Integration on k-1 form

If $\omega$ is a $k-1$ form on a closed $k$-dimensional manifold $M$ then $\int_M d \omega = 0$. I'm looking for a short proof to this problem, would Stokes be helpful?
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### How transversality condition implies that a value is regular?

Currently I am self-learning some manifold theory and just come across concept of functions transverse to submanifolds. It seems that this concept is used a lot for proving regularity of values, but I ...
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### Find a nontrivial bundle of $S^1$ with fibre isomorphic to $\mathbb{R}^n$

Show that such a nontrivial bundle exists for every $n\in\mathbb{N}$. I don't really have any useful ideas here. I'm not sure if there is a general approach I should be taking or if there is just a ...
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### analysis on manifods

Let $M$ be a compact oriented $k+l+1$ dimensional manifold without boundary in $\mathbb R^n$. Let $\omega$ be a $k$-form and let $\eta$ be an $l$-form, both defined in an open set of $\mathbb R^n$ ...
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### Why is $\int_{X} d(\alpha \wedge *\bar{\beta})$ zero on a compact hermitian manifold?

I am reading the book Complex Geometry - An Introduction by Huybrechts. In proving Lemma 3.2.3 that $\partial$ and $\partial^*$ are formal adjoints to each other, he mention that the following ...
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### Specific example of integrating a 1-form over a curve

I was given the following definition in my course but no corresponding examples: Supppose $\gamma:[a,b]\rightarrow{M}$ is a smooth curve and $\omega$ a 1-form on $M$ (so $\omega:M\rightarrow{T^*M}$). ...
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### Counterexample for the density of smooth functions in Sobolev spaces on a manifold

I'm in desperate need for help understanding a counterexample for the assertion that for (appropriate) manifolds $M,N$ the space $C^\infty (M,N)$ is dense in $L^p (M,N)$ if $\dim(M) > p$. (The ...
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### What is concretely a vector field?

Let $M$ be a manifold and $TM=\coprod_{x\in M}T_xM$ be the tangent bundle. By definition, a vector field is an application \begin{align*} X: M&\longrightarrow TM\\ m&\longmapsto X_m\ni T_mM ...
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### What can you say about the $k$-th cohomology group of a closed orientable $n$-manifold for $k = n$ and $k = n-1$?

What can you say about the $k$-th cohomology group of a closed orientable $n$-manifold for: (1) $k = n$, and (2) $k = n - 1$. Poincaré Duality tells us that for $M$ a closed ...
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### Proof formalization help: Given a vector $u$ in $\mathbb{R}^3$ and a compact 2 dimensional manifold $m$, $u$ is normal to the $m$ at 2 points.

Proof formalization help: Given a vector $u$ of Euclidean length $1$ in $\mathbb{R}^3$ and a compact 2 dimensional manifold $m$, $u$ is normal to the $m$ at at least 2 points. I've thought about the ...
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### Computing relative de Rham cohomology?

Let $f: N \to M$ be a smooth map. Then in Bott and Tu they define a relative de Rham complex $\Omega^{\bullet}(f)$ where $\Omega^k(f) = \Omega^k(M) \oplus \Omega^{k-1}(N)$ with coboundary map given in ...
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### Showing that the set of semi-orthogonal matrices is a $C^\infty$ submanifold

For $k, n \in \mathbb{N}$ with $k ≤ n$, we define $$S_{n, k} = \{X \in \mathbb{R}^{n \times k}: X^t X = I_k\}$$ where $I_k$ is the identity matrix of rank $k$. I want to prove that $S_{n, k}$ is a ...
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### Under which additional hypothesis are open maps locally injective

Recollection of basic definitions: We recall the basic definitions that a continous map of topological spaces $f : X \to Y$ is open if $f(U)$ is an open subset of $Y$ whenever $U$ is an open subset ...
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### Is the cuspidal cubic $\{y^2 = x^3\} \subset \Bbb R^2$ not smooth?

Cuspidal cubic $y^2=x^3$ in $\Bbb R^2$ "seems to be not smooth" intuitively because its pictured graph has a cusp at the origin. But I read from book that it is a smooth manifold. I feel so confused. ...
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### explaination of the metric tensor on another manifold?

In skew -product decomposition the following features are observed :- 1.the Riemannian Manifold $(M,g)$ has a product form of $$M=R\times \Theta$$ Where $\Theta ,R$ are connected $C^\infty$ ...
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### the definition of relative Thom spaces

I find the definition of Thom space on the book Characteristic classes, J.W. Milnor, J.D. Stasheff, page 205: For a vector bundle $\xi$ with a Riemannian metric over a $CW$-complex $B$, let $A$ ...
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### Two proof of Petersen's 'Manifold'

Picture below is from the 5 page of Petersen's Manifold. First, why diffeomorphism is forced to be a lieanr isomorphism? The define of diffeomorphism accords to Wiki. Second , what space the point ...
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### Poincaré Duality in Middle Dimension

I am reading a paper that states the following theorem without proof: Poincaré duality in middle dimension: Let $M$ be a connected oriented manifold of even dimension $2d$. Then the cup product ...
Let $\pi: V \to M$ be a smooth $n$-dimensional vector bundle over $M$. Are the spaces of differential forms $\Omega^i(V)$, $\Omega^i(V^*)$ noncanonically isomorphic? If so, how do I see this? Is there ...
### Left-invariant vector fields on the circle $S^1$
I'm trying to find the left-invariant vector fields on the circle $S^1$. If I understand correctly, $S^1$ is given the group structure of the multiplicative group of complex numbers on the unit ...