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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.

8723 questions
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How to prove figure eight is not a manifold?

Possible Duplicate: A wedge sum of circles without the gluing point is not path connected I know that figure eight is not a manifold because its center has no neighborhood homeomorphic to $\mathbb{R}^n$. But how to prove this strictly?
Gobi
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$M/\Gamma$ is orientable iff the elements of $\Gamma$ are orientation-preserving

The probem is: Suppose $M$ is connected, orientable, smooth manifold and $\Gamma$ is a discrete group acting smoothly, freely, and properly on $M$. We say that the action is orientable-preserving if for each $\gamma \in \Gamma$, the…
Kelson Vieira
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What kind of tools do we have to detect when a manifold is a product of other manifolds?

What sort of tools are out there that can detect when a manifold is a product of other manifolds? For example, comparing the homology of the circle to the torus, the homology of the torus gets more "complicated" as in the betti numbers for $H_1(T)$…
Felix Y.
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Embedding of 3-manifold $S^3/Q_8$ in $\mathbb R^4$

In paper of Hantzsche (1938) there is proof that boundary of tubular neighborhood of $RP^2$ in $R^4$ (denote it $M$) is $S^3/Q_8$. I was trying to read this work but I don't know German, so this is difficult. I Hantzsche's paper there is following…
mmm
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Transformation Rule for a Wedge Product of Covectors

Suppose two sets of covectors on a vector space $V$, $\beta^1,\ldots,\beta^k$ and $\gamma^1,\ldots,\gamma^k$, are related by $$\beta^i=\sum_{i=1}^ka^i_j\gamma^i,\quad i=1,…,k,$$ for a $k\times k$ matrix $A=[a^i_j]$. Show that $$\beta^1…
anon271828
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Definition of manifold

From Wikipedia: The broadest common definition of manifold is a topological space locally homeomorphic to a topological vector space over the reals. A topological manifold is a topological space locally homeomorphic to a Euclidean space. In…
Tim
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Are matrices with determinant zero a manifold?

Consider the set of matrices with determinant zero in $M_n(\mathbb R)$, where $n > 1$. Is it a manifold? In fact, is it even a topological manifold? I would suspect not; but I do not have a proof. If it is indeed not a manifold, is it perhaps a…
Rolanda Hooch
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Diffeomorphic Level sets Of Manifolds

Let $F:M^n \to \mathbb{R} $ be a smooth function admitting only regular values and $(M,g)$ a smooth connected riemannian manifold. I know that the vector field $ \frac{\operatorname{grad}F}{||\operatorname{grad}F||^2} $ defined my means of the…
joshua
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Vector field on riemannian manifold

Let $M$ be a riemannian manifold and $N$ be a submanifold of $M$. Let $v$ be a vector field in $N$. Then $v$ can be covariantly differentiated along $\gamma$ resulting in new field $u$. (Here consider Levi-Civita connection) Because $N$ is embedded…
Alexander Taran
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Functions with zero derivative on manifolds are constant.

This seems obvious, but I'm having trouble carrying through the details. Suppose there is a smooth function $f$ with zero derivative on a manifold $M$ with $n$ connected components. Why is $f$ constant on each connected component? Detailed answers…
Potato
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Four mutually equidistant points on a 2D manifold?

In a certain math joke, the humor comes from the insight that having four people who are each 6ft away from one another is absurd because it violates the Pythagorean theorem. This made me wonder whether such a setup is actually possible in a certain…
user326210
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Help understanding Proof of Fundamental Theorem of Algebra

In the book "Topology from the Differential Viewpoint", Milnor gives a proof of the fundamental theorem of algebra. It goes essentially like this: Consider the stereographic projection $h_{+}$ of $S^2$ onto $C$ from the north pole and the…
user140776
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A "Trivial" Smooth Structure

Let M be a topological manifold of dimension $n$. Suppose that there exists a homeomorphism $$ \phi \colon M \longrightarrow \mathbb{R}^n $$ defined for all $x \in M$. That is, $\phi$ is a continuous bijection with a continuous inverse. Thus, it…
ItsNotObvious
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How to show a level set isn't a regular submanifold

For $F:\mathbb{R}^2\rightarrow \mathbb{R}$ given by $F(x,y)=x^3+xy+y^3$, how do I show that $F^{-1}(0)$ and $F^{-1}(1/27)$ aren't regular submanifolds? I've plotted these on Wolfram alpha: the first one crosses itself at a point (so it's not a…
Alex567
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Tangent space of all real symmetric matrices with a fixed rank

Suppose $M_r$ is the set of all real symmetric matrices of order $n$ with rank $r$. (a) Show that $M_r$ is a submanifold of the space $\mathbb R^{n^2}$. (b) Find the tangent space of $M_r$ at some $A\in M_r$. Suppose $N_G$ is the set of all real…
user87118
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