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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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Show that $ T_{p}M $ is a vector space of dimension $ n $.

Show that $ T_{p}M $ is an $ n $-dimensional vector space. Hint: Given two tangent vectors $ v_1 $ and $ v_2 $ at $ p $ with corresponding curves $ \gamma_1 $ and $ \gamma_2 $, we can “add” the corresponding curves in $\mathbb{R}^n$ and then move…
alireza911
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Is there a countable dense subset?

If a manifold $M$ is $\sigma$-compact, does $M$ possess any countable dense subset?
lotz84
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Are $n$-dimensional manifolds locally homeomorphic to $n$-dimensional Euclidean space $\mathbb{E}^n$ or the more general real space $\mathbb{R}^n$?

Many if not most authors use the term “$n$-dimensional Euclidean space” as synonymous with “$n$-dimensional real space”, $\mathbb{R}^n$. Some, however chose to be more rigorous and use the term Euclidean space and the symbol $\mathbb{E}^n$ to…
user175324
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Does this 'n-transitivity' property of manifolds have a name?

It seems that if I take any n points in a connected manifold of 2 dimensions or more, $M$, $x_i$, and any other points in $M$, $y_i$, I can continuously move every point in $M$ around until $x_1$ is at $y_1$, $x_2$ is at $y_2$, etc. I think this…
Zoe Allen
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Find orientation of manifold

In $\mathbb{R}^3$ consider the the 2-dim manifold $$ M:=\left\{(x,y,z)\in\mathbb{R}^3:z=xy\right\}. $$ Let the orientation of $M$ be in such a manner that in the Point $(0,0,0)$ the vector $e_3=(0,0,1)$ is the positive orientated normal vector.…
user34632
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Ring of germs and its unit

If $O_p$ is the ring of germs of smooth functions at point $p$ and $\alpha\in O_p$ represents $(U,f)$, and the map $I:O_p→R,(U,f)\xrightarrow{}f(p)$ is an $R$-algebra homomorphism. It says that if $I(\alpha)$ is not zero, then $\alpha$ is a unit. I…
hslo
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tangent vector of manifold

According to this Wikipedia site https://en.wikipedia.org/wiki/Tangent_space "We can therefore define a tangent vector as an equivalence class of curves passing through x while being tangent to each other at x." and "... equivalence classes of such…
Ruye
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Partial Derivative on a Manifold

Compute the partial derivative of $f(x,y) = 2x + y^3$ at $a = (x_0,y_0)$. I know this looks easy but the purpose of me asking this question is to see how this question is worked as if it we were just working on a general manifold & needed to use all…
derive
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Showing the inclusion map of a submanifold is an immersion.

In the recent course of studying manifolds by the lecture note provided by Ed Segal of the UCL(http://www.homepages.ucl.ac.uk/~ucaheps/papers/Manifolds%202016.pdf), I encountered a question asking me to show that, for a submanifold Z of X the…
Neophyte
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Taylor's theorem with remainder

In Loring Tu's Introdution to Manifolds pg. 6, it was said that if $f$ is a $C^\infty$ function, then $$g_i(x)=\int^1_0 \frac{\partial f}{\partial x^i}(p+t(x-p))dt$$ is $C^\infty$. Tu used $x$ in two different ways, both as a variable and as a…
TaeNyFan
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Sphere is a topological manifold.

If I want to show that the spere $\mathbb{S}^1$ is a topological manifold. Graphically it's clear that we have a chart $x:\mathbb{R}\rightarrow \mathbb{S}^1$ since we can "cut" $\mathbb{S}^1$ and and form it to a line right? But somehow I can't…
user123234
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Question on the relationships of two and three manifolds

The Question is: Let $W_c = \{ ( x,y,z,w) \in R^4 | xyz = c \}$ and $Y_c = \{ ( x,y,z,w) \in R^4 | xzw = c \}$. For what real numbers $c$ is $Y_c$ a three-manifold? For what pairs $(c1,c2)$ is $W_{c_1} \cap Y_{c_2}$ a two-manifold? I think that we…
Ness
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What is the fundamental shape used for enclosing a 4-dimentional object?

In 2-dimensions, any enclosed space can be approximately represented with line-segments composed of the space between 2 points. In 3-dimensions, any enclosed space can be approximately represented with triangles composed of the space between 3…
Nosajimiki
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Invariance of Dimension for the zero-dimensional case

Currently I'm reading through Lee's Introduction to Topological Manifolds. Although the full proof is not given until much later, he gives the following theorem: Theorem 2.55 (Invariance of Dimension) If $m\neq n$, a nonempty topological space…
Rough_Manifolds
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How to prove that TM at p is isomorphic to R^N

How does one prove this, independently of charts used for a manifold? I'm not sure where to even begin. I've seen people do it with equivalence classes of curves defining the TM but I can't visualize how to do that and then I have a problem with the…
James Shelton
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