# 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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### Convert line parametrization into two equations

Consider the following parametrization on $\mathbb{R}^3$ $$g(t) = (t^2,t\cos(t),t\sin(t))$$ This is a line, and as such can be characterized by two equations. I already found the first one to be ...
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### how to prove that $C^{k}$ map does not depend on choice of the charts

I was reading an article about Manifolds.They have defined a $C^{k}$ function in the following way : Let $M$ and $N$ are two $C^{k}$ manifolds of dimensions $m$ and $n$ respectively.A continuous ...
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### $A\cup \text{Fr}A$ is a manifold

I'm dealing with the following exercise from Spivak's "Calculus on Manifolds": Let $A\subset \mathbb R^n$ be an open set such that the (topological) boundary of $A$, $\text{Fr}A$ is an $n-1$ ...
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### Conditions on characteristic polynomial to define a matrix submanifold.

I'm trying to find conditions on the characteristic polynomial, $p$, of a matrix such that the pre-image of matrices with characteristic polynomial $p$ form a manifold. More precisely, we can write ...
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### Prove that integration of a differential $k$-form is independent of choice of basis

This is Exercise 4 of Section 33 of Munkres' "Analysis on Manifolds" book: (Let $A$ be an open set in $\mathbb{R}^k$.) If $\eta$ is a $k$-form in $\mathbb{R}^k$ and if ...
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### How to define this space? (matrix of coordinates)

We will let $F$ denote an arbitrary field such as the real numbers $R$ or the complex numbers $C$. For any positive integer $n$, the space of all $n$-tuples of elements of $F$ forms an ...
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### Can we recover a compact smooth manifold from its ring of smooth functions?

It is well-known that if $X$ is a reasonably nice topological space (compact Hausdorff, say) then we can recover $X$ from the ring $C(X)$ of continuous functions $X\to\mathbb R$; see this MSE question ...
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### $\mathbb{CP}^2$ realizable as boundary of compact smooth $5$-manifold? [duplicate]

As the title says, can $\mathbb{CP}^2$ be realized as the boundary of a compact smooth $5$-manifold?
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### What does $C^k$ at a single point mean?

I'm reading textbooks on manifolds. I usually see a saying that "function $f$ is $C^k$ at a point $p \in M$". I am confused with this. What does this mean? Furthermore if $f$ is $C^k$ at $p$, is it ...
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### What is the motivation for wanting to remove redundant terms that arise from the wedge product of two multilinear functions?

I am currently working through Loring Tu's "An Introduction to Manifolds," specifically the section in which the exterior algebra of multicovectors is introduced, and I am having trouble grasping the ...
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### On computing the Differential of a Smooth Map

In class, we proved Jacobi's formula for the differential of the determinant using the following formula for the differential of a smooth map $F$ between manifolds $M$ and $N$ D_A F(B) = ...
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### Why is $\phi^* g = g$ a PDE for a pseudo-Riemannian metric $g$ on a manifold?

Given a (locally trivial) bundle $\pi: E \to M$ a PDE of order $k$ is usually defined to be a submanifold of the jet-bundel $J^k(E)$. Now assume $E = M \times M$ and $\pi$ is the projection on the ...
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### What is the skew-symmetric part of the covariant derivative of a one-form?

This is a followup question to here. Let $E \to M$ be a vector bundle with connection $D := \nabla$. Extend $D$ to $E^*$ and $\text{Hom}(E, E)$. Let $E = TM$ here, and suppose that the torsion is ...
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### Which homology groups of a closed orientable 6-manifold can be isomorphic to $\mathbb{Z}^3$?

List all $i$ for which there is a closed orientable $6$-manifold $M$ with $H_i(M) =\mathbb{Z}\oplus\mathbb{Z}\oplus\mathbb{Z}$ I am working on an old exam problem and this one stumped me. ...
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### Why is the image of the implicit function in the implicit function theorem not open?

We have a continuously differentiable function $f$ from $\mathbb{R}^{n+m}$ to $\mathbb{R}^n$, and we find a continuously differentiable function $g$ which maps points from $\mathbb{R}^m$ into ...
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### Difference between Grassmann and Projective space?

I recently read the first few chapters of Pyramid Algorithms: A Dynamic Programming Approach to Curves and Surfaces for Geometric Modeling and I am confused about ...
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### Question about connections on the dual bundle.

Let $E \to M$ be a vector bundle with connection $\nabla$. Extend $\nabla$ to $E^*$ and $E^* \otimes E$ in the regular fashion. Is $\text{Id} \in E^* \otimes E$ necessarily parallel?
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### Quadrature over a (smooth, compact, convex, etc.) Riemannian manifold

Problem setting Consider three points on the surface of the earth (which I want to assume to be a perfect ellipsoid here) that are pairwise sufficiently close for unique geodesics to be found between ...
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### When and where to check the formal definition of a manifold

In most texts on GR we are first introduced to a formal and rigorous definition of a manifold. We then learn the point that in GR "any coordinate system" might be used for the 4D spacetime metric. ...
### Every $\mathcal{C}^1$ manifold can be made smooth?
I heard of a theorem saying that each $\mathcal{C}^k$-manifold with $k\geq 1$ can be made into a smooth manifold, i.e. $\mathcal{C}^{\infty}$ (by restriction of the atlas). However, I cannot find ...