# 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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### 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 $R$-...
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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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Let $M$ be a manifold of dimension $d$ and let $\mathsf{Disk}_{/M}$ be the category of open subsets of $M$ that are diffeomorphic to $\mathbb{R}^d$ with morphisms given by inclusions. Let $\mathrm{B} ... 1answer 25 views ### Calculating the second fundamental form of surfaces I am asked to prove that for a surface in$\mathbb{R}^3$with local coordinates in a chart, u,v, the coefficients of the second fundamental form can be calculated as follows: eg.: (first entry in II) ... 1answer 38 views ### Integral of Differential 1-form Let$\omega$be the closed$1$-form$\omega = \frac{xdy - ydx}{x^2 + y^2}$and let$S$be the unit circle and let$C = \{(x,y) : (x-3)^2 +y^2 = 1\}$. I'm trying to find$\int _S \omega$and$\int ...
Problem: Calculate $\int_S dx \wedge dy + dy \wedge dz$, where $S$ is the surface given by $S = \{(x,y,z) : x = z^2 +y^2 -1, x < 0\}$. Wikipedia says: Let  \omega=f_{z}\, \mathrm dx \wedge \...