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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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How to calculate the tangent space?

first of all, what is the difference between the tangent space and the tangent plane? I tried to find the tangent space of the hyberpoloid $$x^2 +y^2 -z^2 =a$$ , $$a>0 $$ at the point $$(\sqrt{a},0,0)$$ in this way: $$f(x,y,z)=x^2 +y^2…
letisya
  • 111
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Why this manifold can not be embbeded in 3d Euclidean space?

Consider the following system of 6 equations in 9 variables $x_1, x_2,x_3,...x_9$ $ x_1 ^ 2 + x_2 ^ 2 + x_3 ^ 2 + = 1 $ $ x_4 ^ 2 + x_5 ^ 2 + x_6 ^ 2 + = 1 $ $ x_7 ^ 2 + x_8 ^ 2 + x_9 ^ 2 + = 1 $ $ x_1 x_4 + x_2 x_6 + x_3 x_7 = 0 $ $ x_1 x_7 + x_2…
Hooman Safi
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Is a tangent vector field on $S^n$ always continuous?

Let's say we have an arbitrary tangent vector field on $S^n$. Can just say that the tangent vector field is continuous? Edit: I saw this in Hatcher's book that "$S^n$ has a continuous field of tangent vectors iff $n$ is odd" and wondered if…
Emily
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Manifolds and actions: a counterexample

Let $X$ be an $m$-dimensional simply connected manifold that is also a G-space. If the action is properly discontinuous, is $X/G$ necessarily a manifold? Do you have a hint for this one?
absalon
  • 347
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How to find the tangent space of $O(n)$ by considering $O(n)$ as the pre-image of the map $A \mapsto AA^T$ at identity?

Why is the tangent space of $O(n)$ at $H$ equal to $T_H O(n) = \{ M \in \mathbb{M} ( n, \mathbb{R} ): (DF(H))(M) = 0 \}$, where $$F: \mathbb{M} ( n, \mathbb{R} ) \cong \mathbb{R}^{n^2} \to \mathbb{Sym}(n)\cong \mathbb{R}^{\frac{n(n+1)}{2}}$$ is…
math.n00b
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Smooth Manifolds- Question From Lee's Book

I really need your help in understanding the following statement in the proof of the extension lemma in Lee's book: Let $A \subseteq M^n $ be a closed submanifold of dimension $k$ , and let $F:A \to \mathbb{R} $ be a smooth function. We want to…
joshua
  • 1,269
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Charts/transition charts of the $\mathbb{CP}^3$ tangent bundle

I would like to explicitly compute the charts and transition charts for the tangent bundle of $\mathbb{CP}^3$. I know the charts of $\mathbb{CP}^3$ are $\phi_i: U_i=\{[z_0,z_1,z_2,z_3]; z_i \neq 0\}\rightarrow…
Bruno Suzuki
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Understanding the equivariant rank theorem

The statement: Let $M$ and $N$ be smooth manifolds, and let $G$ be a Lie group. Suppose $F:M\rightarrow N$ is a smooth map that is equivariant with respect to a transitive smooth $G$-action on $M$, and any smooth $G$-action on $N$. Then $F$ has…
Sid Raval
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Neighbourhood in a manifold is open

I'm trying to solve a problem in Spivak's A comprehensive introduction to differential geometry. Here, the definition of a manifold is the next A metric space $X$ is said to be a manifold if every point in $X$ has a neighbourhood homeomorphic to…
Brandon
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Glueing cubes to manifolds with corner

I am interested in proposition 3.7 in Salvatore's `Configuration spaces with summable labels'. The result states that the bar construction on the Fulton-Macpherson operad is isomorphic to the FM-operad itself. In his proof, the author claims that if…
Dean Barber
  • 176
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Question on exact $n$ form with compact support

I've encountered with the following problem: Consider the map $$\int: \Omega_c^n(\Bbb R^n)\to\Bbb R$$ $$\alpha(x)dx^1\land...\land dx^n\mapsto \int_{\Bbb R^n}\alpha(x)dx^1\land...\land dx^n$$ It's trivial that this map$\int$ is a homomorphism,here…
C Weid
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A question about tangent subbundle

Let $x$ be a manifold, $E$ is a subbundle of $TX$ , my question is : Can you give example such that vector fields $\xi ,\eta$ lie in $E$,but bracket $[\xi ,\eta]$ does not lie in $E$ in some point of $x$.
gilliatt
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Pushforward of the tangent space at a point of a submanifold

Let $\iota : S \hookrightarrow M$ be an inclusion that serves as an injective immersion between real manifolds of dimension $k$ and $n$, respectively. Fix $p \in S$. Then we have a linear embedding $\iota_{*, p} : T_{p}S \hookrightarrow T_{p}M$, and…
user123454321
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Can a Sphere be Flattened?

We consider 2D metric: $ds^2={d\theta}^2+\sin^2(\theta)d{\phi}^2$ (1) Is it possible to transform the above metric to the form: $ds^2=dx^2+dy^2$ (2) Let's check: Initially we write equation (2) in the form: $ds'^2=dx^2+dy^2$ (3) For relations (1)…
Anamitra Palit
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Typo in Spivak Calculus on Manifolds?

In the proof of theorem 5.2 of Spivak's calculus on manifolds, he invokes the inverse function theorem for a function g that he defines there without ever assuming that it was continuously differentiable. Is this a typo?
user68432
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