# 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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### Is an open subset $U \subset \mathbf{R}^{n}$ diffeomorphic to the product $U' \times \mathbf{R}$ with $U' \subset \mathbf{R}^{n - 1}$ open?

I'm trying to prove that $U$ is diffeomorphic to the product of some open subset $U' \subset \mathbf{R}^{n}$ with $\mathbf{R}$, $U' \times \mathbf{R}$. I received the hint that this set admits a ...
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### Trying to prove that $TM$ is a manifold: Is this function an homeomorphism?

I am trying to prove that if $M$ is a $k$-manifold in $\mathbb R^n$, then $TM=\{(p, v): p \in M, v \in T_pM\}$ is a manifold. Here, $T_pM$ is defined as a subset of $\mathbb R^n$. I know that ...
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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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### Symplectic form on $T^ ∗X$

If $\mu$ is any closed 2-form on a manifold $X$, then $d\alpha_X + π^∗\mu$ is also a symplectic form on $T^ ∗X$, where $\alpha_X$ is tautological one-form and $\pi:T^*X\to X$ is projection. In fact, ...
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### Is there an atlas of Kirby diagrams of 4-manifolds?

We've defined an invariant of 4-manifolds (article) in terms of Kirby diagrams and I'm looking for a lot of manifolds to test it on. Now I'm not a big differential topologist myself, and coming up ...
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### Explicit form of the exponential map

I am stuck by the following problem. Let $g = 1/y^2 (dx \otimes dx + dy \otimes dy)$ be a Riemannian metric on the half-plane $S = \mathbb{R}^2_{y > 0}$. What is the explicit form of the ...
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### Oriented atlas on a circle

I'm trying to find an oriented atlas on the circle $S^1$, i.e., I want to find an atlas for $S^1$ such that for any two overlapping charts $(U,s)$ and $(V,t)$ of the atlas, the derivative $d s/d t$...
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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 ...
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 $\mathbb R^n$ ...
### 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 ...