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Questions tagged [manifolds]

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

In mathematics, a manifold of dimension $n$ is a topological space that near each point resembles $n$-dimensional Euclidean space. More precisely, each point of an $n$-dimensional manifold has a neighbourhood that is homeomorphic to the Euclidean space of dimension $n$. Lines and circles, but not figure eights, are one-dimensional manifolds. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, which can all be realized in three dimensions, but also the Klein bottle and real projective plane which cannot.

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Pulling back forms computation

Let $f:\mathbb{R}^2 \to \mathbb{R}^2$ be a function mapping $$(x, y) \to \left(e^{2x}, xy\right).$$ How do you compute pulling back form of a 2-form $$\alpha(x, y)=xy(dxdy),$$ in other words, $f^*(\alpha)$?
jfcjohn
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Differential manifold of dimension $m^2$

How could I prove that $\mathrm{GL}(m, \mathbb{R})$ is a differential manifold of dimension $m^2$? $\mathrm{GL}(m, \mathbb{R})$ is the set of all non-singular $m\times m$ matrices in $\mathbb{R}$ Thank you
Cherax
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Bases for the spaces$\mathcal{V}^1(X)$ and $\Omega^1(X)$

For precise definitions of the spaces referenced below, please refer to this question. For $X \subset \mathbb{R}^n$, I understand that the $n$-dimensional tangent space $T_pX$ has a natural/canonical basis $((e_1)_p, \dots, (e_n)_p)$ where each…
ItsNotObvious
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Explain why this set is not a differentiable manifold

I want to figure out why the set of zeros of the function $g:\mathbb{R}^{2} \to \mathbb{R}$ defined as $g(x,y) = x^2 - y^2$ is not a differentiable manifold. So what I want to use is the following result: Let $A \subset \mathbb{R} $ open, $p
user162343
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Quotiented manifold homeomorphic to a complex projective space?

I define an action on $\mathbb{C}-0 × \mathbb{C^2}-(0,0)$ by $(x,y,z) \mapsto ((1/a)x,ay,az)$ when $a$ is a non zero complex number, I get a manifold by quotienting. Taking element from this quotiented manifold and quotienting back in the…
Nestor
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Discrete singularities of $C^k$-functions.

I'm stacked in the following problem: suppose $f:M\rightarrow N$ is a $C^k$ map between $C^k$-manifolds, such that $\dim M=\dim N=n>1$; if the singularities of $f$ are isolated, then the map is open. I've tried to use the fact that in the complement…
Renan Mezabarba
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How to show that a subset of $\mathbb{R}^2$ is a closed one-dimensional submanifold?

I'm trying to solve the following problem: For $c \in \mathbb{R} \setminus \{ 0 \}$, let $$C = \{(x,y) \mid x^3 + xy + y^3 = c \} \subset \mathbb{R}^2.$$ Show that for $c \neq 1/27$ the set $C$ is a closed one-dimensional submanifold of…
Maethor
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How to show that a map $F:M\to N$ between manifolds is smooth if $f\circ F$ is smooth for smooth $f:N\to R$.

Let $M,N$ be smooth manifolds (in the sense that all of their charts have partial derivatives of all orders). Let $F$ be a map from $M$ to $N$. Suppose that for any smooth $f: N\to \mathbb{R}$, $f\circ F$ is a smooth function. Then I would like to…
user21725
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Equivariant versions of Gram Schmidt, references needed

I am looking, without luck so far, for references for an equivariant Gram-Schmidt. Specifically, Let $P$ be the permutation matrices, and $D$ the diagonal matrices with entries in $\mathbb R^\times$. Then $P,D$ have the obvious right actions on…
Andrew Marshall
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Sampling on Manifolds

Let $M$ be a compact manifold of unknown dimension embedded in $\mathbb{R}^n$ for some natural $n$. Let $S=\{x_1, x_2,..., x_k\}$ be a uniform random sample obtained from $M$. Let $\Gamma(S)$ be the set of all geodesics between points in $S$ on…
Joseph Zambrano
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Interior of a compact manifold with boundary is compact

In the context of manifold with boundary, closed manifold, compact manifold I have the following question in my mind : Let $M$ be a compact manifold with non-empty boundary $\partial M$. Then $\operatorname{int}( M) =M-\partial M$ is a manifold…
Bingo
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Relation between Map and Dimension

I am curious about two questions below Let $M$, $N$ be two topological manifold. If $\dim M>\dim N$, is there exist an injective continuous map $f: M\rightarrow N$? If $\dim M<\dim N$, is there exist an surjective continuous map $f: M\rightarrow…
gaoxinge
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what is the basis for $A_3(T_p(\mathbb{R^4}))$.

Let $x^1,x^2,x^3,x^4 $ be the coordinates on $\mathbb{R^4}$ and $p$ a point in $\mathbb{R^4}$. Write down a basis for the vector space $A_3(T_p(\mathbb{R^4}))$. I have no idea what is the basis for $k$- linear alternating spaces over vector spaces.…
Bhaskar Vashishth
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How to prove a set is a submanifolds or not

I am studying about differentiable manifolds. My professor give me an example show that graph of a continuous function is a submanifold, but image of its is not in general. $$f: \mathbb{R} \longrightarrow \mathbb{R}^2 $$ $$f(x)=(x^2,x^3)$$ $gr(f)=\{…
user69833
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Hopf map is continuous

Consider map $h:\mathbb{S}^{3}\to \mathbb{S}^{2}$ defined as $h(a,b)=(a\bar{b}+b\bar{a},ib\bar{a}-ia\bar{b},|a|^{2}-|b|^{2})$ Does it just follow by seeing h as map from $\mathbb{C}^{4}$ to $\mathbb{R}^{3}$ or am I missing sth, and it follows from…
TKM
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