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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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The orthogonal group has a unique manifold structure

We proved the following theorem : Assume that $\psi:M^c\longrightarrow N^d$ is $C^{\infty}$. Let $n\in N$, $P=\psi^{-1}(n)$ be non-empty, and let $d\psi:M_m\longrightarrow N_{\psi(m)}$ is surjective for all $m\in P$. Then $P$ has a unique manifold…
gradstudent
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General linear group on manifold

Knowing that $\mathrm{O}(n,\mathbb{R})$ is a closed submanifold (of the general linear group) and that $\mathrm{SO}(n,\mathbb{R})$ is one of its subgroups with the same dimension, is there a quick way (possibly using only basic arguments, i.e.…
Jean
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Product Manifolds and Tangent spaces

Let $M\subset{E^{n}}$ be an r manifold and $N\subset{E^{m}}$ be an s manifold. Regarding $E^{m+n}$ as the Cartesian product $E^{n}\times{E^{m}}$, show that $M\times{N}$ is an (r+s)manifold. Show that the tangent space at a point of $M\times{N}$ is…
Robert
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A Manifold Contained in Another

QUESTION: Let $M$ be a $k$-manifold-without-boundary in $\mathbb R^n$ and $N$ be another manifold-without-boundary in $\mathbb R^n$ such that $M\subseteq N$. Assume that there exists a point $\mathbf p\in M$ such that for each open set $U$ in…
caffeinemachine
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Torus, manifolds

I have some trouble with the following questions: $\mathbb{R}^3$ has standard coördinates $(x, y, z)$. Regard in the plane $x=0$ the circle with centre $(x,y,z) = (0,0,b)$ and radius $a$, $0
Leslie
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Definition of Manifold's Orientation

I am reading the book of manifold. And I find there are many definitions about one object, such as orientation, Euler character and degree of map. I am confused with the conception of orientation. This may be related to…
gaoxinge
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Manifold in $\mathbb{R}^2$

Is the set $\left\lbrace (x,y) \in \mathbb{R}^2 \mid x^2 = y^2 \right\rbrace $ a manifold in $\mathbb{R}^2 $? I know I could use the level set theorem if I had $\left\lbrace x \in \mathbb{R}^2 \mid x_1^2 = x_2^2=c \right\rbrace $, provided $ c…
Kerry H
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Manifolds and Topological Spaces

from my understanding of manifolds they are structures defined on topological spaces. So if M is a manifold defined on a topological space $(X,\tau)$ and $X\subseteq\mathbb R^3$, does this mean $M$ is a $3$-manifold? If so does this generalize to…
Christian lapointe
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Computing tangent space for quadric

What is the tangent space to the quadric $x_1^2+x_2^2+\ldots+x_{n-1}^2=x_n^2$ at the point $p=(1,0,\ldots,0,1)$? The definition of a tangent space that I know is based on the fact that we have a manifold $X$, and then finding a map $f$ from a…
Mika H.
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Critical Point and the Orthogonality of the Gradient to the Tangent Space to a Manifold

Let $\phi:M\rightarrow \mathbb{R}$ be smooth, M be a k-dimensional submanifold, and $F:U \rightarrow M$ be the inverse map of a local coordinate near $p \in M$ where $ U \subseteq \mathbb{R}^n$. How can I show that $\phi \circ F$ has a critical…
user14503
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Does the inverse mapping theorem in the setting of subsets of $\mathbb{R^n}$ imply the inverse mapping theorem in the setting of manifolds?

Does the inverse mapping theorem in the setting of subsets of $\mathbb{R^n}$ imply the inverse mapping theorem in the setting of manifolds? Or does one have to plunge back into the original proof and make suitable adjustments there? I'm not seeing…
user93826
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Tangent vectors on manifold, different definitions

I have problem with understanding difference between definitions of tangent vectors. Can you explain which definition of tangent vector is the best? (in case of finite dimensional manifold) I know 3 definitions, germs, curves, operators (they are…
user109301
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Differential of a $C^\infty$ function from a manifold $M$ to $\mathbb{R}$

I am studying differentiable manifolds from Warner. In the book, the differential is defined as follows. If $\psi:M\longrightarrow N$ is $C^\infty$, and if $m\in M$, then the differential of $\psi$ at $m$ is the linear map $d\psi:M_m\longrightarrow…
gradstudent
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why f(M) is sub manifold

Let map f of M into N be an injective immersion. show taht if M is compact then f(M) is submanifold of N.
sara
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Subset of non-units of germs of smooth functions at $x$ is an ideal

For a manifold $X$ with a point $x \in X$ define the ring of germs of the smooth functions at $x$ to be $C^{\infty}_x(X)=C^{\infty}(X)/\sim$ where $f_1 \sim f_2 \iff \exists U \in \tau_X.f_1|U=f_2|U$ where $\tau_X$ is a topology on $X$. I assume the…
Dávid Natingga
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