Questions tagged [manifolds]
For questions on manifolds of dimension $n$, a topological space that near each point resembles $n$-dimensional Euclidean space.
9,113 questions
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Integration of differential $k$-form along continuous singular $k$-chain
Suppose that $M$ is a smooth manifold, and let $\omega$ be a fixed differential $k$-form.
Let $C$ be a singular $k$-chain.
My goal is to define $\int_C \omega$, which is usually impossible because $C$ ...
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Clarification on the definition of $C^r$ maximal atlas via unions of equivalent atlases. ("Basic Manifold Theory" by Yukio Matsumoto.)
In "Basic Manifold Theory" by Yukio Matsumoto, a $C^r$ maximal atlas is defined as follows:
Let $S$ be a $C^r$ atlas on a topological manifold $M$. Then the union of all $C^r$ atlases ...
- 9,896
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How do I prove the equivalence between the derivation definition of a tangent space and the tangent vector definition?
One way of defining the tangent space of an $n$-manifold $(M,C)$ at a point $p$ is by a derivation. A derivation $D:S(p)\rightarrow
\mathbb{R}$ is something that satisfies
$$
\begin{align}
D(f+g)&=...
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Are Taylor / Laplace approximations possible also using gradients defined for manifolds like $S^1$?
For example, consider the sphere $S^2$, some point $x_0$ on the sphere and a differentiable function $f: S^2 \to \mathbb{R}$.
I know there's a definition in differential geometry for the gradient at $...
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Mapping of Geodesic Equation onto a Euclidean Plane
The Question:
In the 2-space with line element
$$ds^{2}=\frac{dr^{2}+r^{2}d\theta^{2}}{r^{2}-a^{2}}-\frac{r^{2}dr^{2}}{(r^{2}-a^{2})^{2}},$$
where $r > a$, show that the differential equation for ...
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Existence of smooth bump function without using partition of unity
This problem is an exercise of my course.
My professor does not put second-countability on the definition of smooth manifold.(Our "smooth manifold" is a locally Euclidean Hausdorff space ...
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Why is it so difficult to answer such a basic question in this field? ("Basic Manifold Theory" by Yukio Matsumoto.)
I am reading "Basic Manifold Theory" by Yukio Matsumoto:
Even if a differential structure exists on a topological manifold, it is not necessarily unique. The first discovery of an example ...
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How to rotate a tangent vector at a point on a Riemannian manifold
TL;DR: How can I, in the same way I can derive the inner product formula from a metric tensor (on a Riemannian manifold), derive a full rotation formula from a metric tensor?
Hello, I've been self-...
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A mistake in Theorem 25.4 in Munkres' “Analysis on Manifolds” (Computing integrals on manifolds in a piecewise manner.)
Here is the context.
The areas needing modification are indicated by red markings with sequential numbering in the image above.
(1) “Then $\int_{M}f\mathrm{d}V=\sum_{i=1}^{N}[\int_{A_{i}}(f\circ\...
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A Serious Critique of “Theorem 25.4” in Munkres' “Analysis on Manifolds” (Computing integrals on manifolds in a piecewise manner) [closed]
Here is the context. (The definition of measure zero in a compact k-manifold and its property, and Theorem 25.4)
In Munkres' treatment of the above content, the conclusions are rather abrupt, making ...
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Proving a property holds on a meager set
Let $\Theta \subset \mathbb{R}^2$ be an open, bounded set with a representative element $(\alpha,\beta)$.
Let $\mathcal{X} = C^2(\Theta)$ be a function space endowed with the Whitney $C^2$ topology. ...
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Every alternating form of $\mathcal{A}^n(V)$ is given by its value on a basis times a determinant [duplicate]
Let $\alpha\in \mathcal{A}^n(V)$ is an alternating form, where $n=\text{dim}(V)$ and $V$ has basis $w_1,\dots, w_n$. Then how can I show that $\alpha(v_1,\dots,v_n)=det(A)\alpha(w_1,\dots, w_n)$, ...
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Does every manifold topology on a cartesian product arise as a product of manifold topologies?
Let $X$ and $Y$ be arbitrary sets. It is fairly well known that given a topology $\theta$ on $X \times Y$, there may not exist topologies $\tau_1$ on $X$ and $\tau_2$ on $Y$ such that $(X\times Y,\...
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The intersection of a planar curve and a box.
Let $M\subseteq \mathbb{R}^2$ be a $1$-dimensional, connected, embedded submanifold of $\mathbb{R}^2$ without boundary, $B:=[a,b]\times [c,d]$ be a box. What can we say about $M\cap B$ (provided that ...
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Doubts about extensions of “class $C^{r}$” and “$D\alpha(\mathbf{x})$” in Munkres “Analysis on Manifolds”
(i) Extend the concept of “class $C^{r}$”.
Definition. Let $S$ be a subset of $\mathbf{R}^{k}$. Let $f\colon S\rightarrow\mathbf{R}^{n}$. We say that $f$ is of class $C^{r}$ on $S$ if $f$ may be ...
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