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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Surfaces with the same constant Gaussian curvatures are locally isometric?

Question: I want to know whether surfaces with the same constant Gaussian curvatures are locally isometric or not. If the answer is yes, why? Definitions: Gaussian curvature of a surface at some point is the product of the eigenvalues of the shape…
user87128
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Oriented 3-manifold fibering over two different surfaces?

In a previous post I asked if a closed connected 3-manifold could have fibrations over surfaces $\Sigma_1$ and $\Sigma_2$ with $\Sigma_1 \ncong \Sigma_2$. I got the great example of $K \times S^1$ where $K$ is the Klein bottle to show that this can…
user101010
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Manifolds with diffeomorphic boundaries

Suppose $X$, $Y$ are two $2n$-dimensional manifolds with handles of index $0$ and $n$. In particular, $X$ and $Y$ have boundary. Suppose that $X$, $Y$ have isomorphic homology and intersection form and that their boundaries are diffeomorphic. Does…
user39598
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Nontrivial map from manifold to real projective space

I am doing now old topology quals from Maryland University, and there is a following problem: If $M$ is compact smooth connected m-manifold, show that there exists non null-homotopic continuous map from $M$ to $\mathbb{R}P^n$, assuming that $m=n$…
Igor Sikora
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Radius of convergence is the point at which the function ceases to be analytic: false for $k \neq \mathbb{C}$?

Theorem: Let $U$ be an open set in $\mathbb{C}$, $f$ an analytic function on $U$, and $z_0 \in U$. Then $f$ has a power series expansion centered at $z_0$. If $r$ is a positive real number, and $U$ contains the disc of radius $r$ centered at…
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Proving that an essential surface in a 3-manifold must be separating under $\pi_1$ assumptions.

I'm trying to justify the following claim I've encountered in a paper but am having no luck. Here's the setup. Let $F$ be a connected, essential surface (possibly with boundary) in a compact, connected, orientable, irreducible 3-manifold $M$…
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Why is $SU(n)/SU(n-1)$ the $2n-1$-sphere?

I am looking at Fomenko, Fuchs' book on "Homotopical Topology" and they claim that we have the isomorphism $$ SU(n)/SU(n-1) \cong S^{2n-1} $$ Why is this true? Here is what I have so far: If I have a matrix $A \in SU(n-1)$, then we can embed…
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Alexander's horned sphere as a sub-manifold

I have some questions concerning Alexander's horned sphere. The horned sphere is an embedding (via some continuous injection $f$) of the usual 2-sphere $S^2$ into $\Bbb R^3$ in such a way that the unbounded part of $\Bbb R^3\setminus f(S^2)$ is no…
M. Winter
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Isometry group of $SL(2,\mathbb{R})$

What is the cleanest way to see that the $SL(2,\mathbb{R})$ group manifold has the isometry group $SL(2,\mathbb{R})_{L} \times SL(2,\mathbb{R})_{R}$? Please be gentle with your answers - I am a physicist by training! My main motivation behind this…
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Proof Of Courant's Nodal Domain Theorem

I ask about proof of Courant's nodal domain theorem. Let $M$ be a Riemannian manifold. Let $0\le\lambda_1 \le \lambda_2 \le \cdots$ be eigenvalues of $M$, and $\{\phi_1,\phi_2,\cdots$} be a complete orthonormal basis of $L^2(M)$ such that $\phi_j$…
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Showing something isn't a manifold

So I'm following some notes that are introducing manifolds with pretty minimal prerequisites. What I want to do is show where the image of $\phi: \mathbb{R}\rightarrow \mathbb{R^2}$ $t\mapsto (t-\sin(t),1-\cos(t))$ isn't a manifold. Since this is…
AsinglePANCAKE
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Example of a space which is Hausdorff and locally Euclidean but not paracompact?

On page 330 of John G. Ratcliffe's text on hyperbolic manifolds, he defines a manifold $M$ to be a locally Euclidean Hausdorff space. By locally Euclidean, he means as usual that for each $x \in M$ there is an open neighbour $U$ of $x$ which is…
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De Rham differential is a derivation of wedge product?

Let $M$ be a smooth manifold and $k$ a positive integer. Let us define $$\Omega^k(M)=\left\{\begin{array}{lcl}C^\infty(M) & \textrm{if}& k=0\\ \mathsf{Alt}^k_{C^\infty(M)}(\mathfrak{X}(M), C^\infty(M)) & \textrm{if} & k>0 \end{array}\right.$$ where…
PtF
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Two smooth bounded connected domains in $\Bbb R^d$ with the same boundary are identical

Let $\Omega_1,\Omega_2\subset\Bbb R^d$ be two connected opens such that $\overline{\Omega_1}$ and $\overline{\Omega_2}$ are smooth bounded connected manifolds with boundary. Then I suspect that if $\partial\Omega_1=\partial\Omega_2$, then…
Dominic Wynter
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Extending a Set of Linearly Independent Vector Fields to a Basis

My question is this. Suppose we are given some smooth vector fields $X_1, X_2,..., X_k$ which are linearly independent at all points in a neighborhood $U$ (EDIT: diffeomorphic to a ball) of $R^n$. Do there necessarily exist smooth vector fields…
Cass
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