For questions on manifolds of dimension $n$, a topological space that near each point resembles $n$-dimensional Euclidean space.

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3answers
184 views

Why $\mathbb{RP}^2$ can not be embedded to $\mathbb{R}^3$?

Is there any answer of this question around basic theory of differentiable manifolds?
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2answers
142 views

$\mathbb{S}^2$ as a fibre bundle

I know, by the Hopf fibration, that $\mathbb{S}^3$ is an $\mathbb{S}^1$-fibre bundle over $\mathbb{S}^2$. Can $\mathbb{S}^2$ be an $\mathbb{S}^1$-fibre bundle over some manifold $M$?
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2answers
649 views

Any example of manifold without global trivialization of tangent bundle

It is said for most manifolds, there does not exist a global trivialization of the tangent bundle. I am not quite clear about it. The tangent bundle is defined as $$TM=\bigsqcup_{p\in M}T_PM$$ So is ...
7
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3answers
255 views

Algebraic varieties in $\mathbb{C}^n$ cannot have interior points

I know that the zero-set of a non-zero polynomial in $\mathbb{C}[x_1,...,x_n]$ can not have interior points, but I'm trying to find a proof that doesn't require a knowledge of complex analysis like ...
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2answers
937 views

map of arbitrary degree from compact oriented manifold into sphere

This is a question from a qualifying exam. Let $X$ be a compact, oriented $n$-dimensional manifold. Show that for any $k \in \mathbb{Z}$, there exists a continuous map $f: X \to S^n$ of degree $k$. I ...
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2answers
786 views

Manifold interpretation of Navier-Stokes equations

I am wondering about particle trajectories for solutions of the Navier-Stokes equation. Is it possible that there is a Manifold $M$ for which fluid particles move along geodesic's or "straight lines ...
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2answers
224 views

Smooth surfaces that isn't the zero-set of $f(x,y,z)$

The zero-set of any smooth function $f(x,y,z)$ with a non-vanishing gradient is a smooth surface. I was wondering if the reverse is true: is every smooth surface in $E^3$ the zero-set of some smooth ...
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3answers
187 views

let $\xi$ be an arbitrary vector bundle. Is $\xi\otimes\xi$ always orientable?

Let $\xi=(E,p,B)$ be a line bundle (not nec. orientable). Then the tensor product $\xi\otimes \xi$ is orientable. I obtain this by choosing $b\in U\cap V$, $U,V$ open in $B$ such that ...
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2answers
473 views

The Heisenberg manifold

I am interested in the Heisenberg manifold, which is the quotient of the real Heisenberg group by the discrete Heisenberg (sub)group. It is a $3$-manifold which may be viewed as a circle bundle over ...
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2answers
917 views

Manifold with different differential structure but diffeomorphic

I'm new to differential geometry and reading Lee's book Manifold and Differential Geometry. In the first chapter, he mentioned the following two maps on $\mathbb{R}^n$: (1) $id: (x_1,x_2\cdots x_n) ...
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3answers
597 views

Prerequisite for Petersen's Riemannian Geometry

A professor recently told me that if I can cover the chapters on curvature in Petersen's Riemannian geometry book (linked here) within the next few months then I can work on something with him. ...
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3answers
495 views

Why are differential forms more important than symmetric tensors?

In differential geometry, differential forms are totally anti-symmetric tensors and play an important role. I am led to wonder why do we not study totally symmetric tensors as much as forms. What ...
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2answers
537 views

Each point of $M$ has a smooth coordinate neighborhood in which the coordinate frame is orthonormal iff $g$ is flat

Let $(M,g)$ be a Riemannian manifold. Then I want to show that these are equivalent: (i) Each point of $M$ has a smooth coordinate neighborhood in which the coordinate frame is orthonormal. ...
7
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1answer
114 views

Cancellation in topological product

I was wondering whether $M\times \mathbb{R}$ is homeomorphic to $N\times \mathbb{R}$ implies $M$ is homeomorphic to $N$, where let us say $M,N$ are smooth manifolds. (They are certainly homotopy ...
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2answers
178 views

How do manifolds have enough structure to do calculus?

I am referring, of course, to to differentiable manifolds. I've seen a few different definitions. The one I like best is the one which says it's a topological space such that every point has a ...
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2answers
742 views

A Banach Manifold with a Riemannian Metric?

Given an infinite dimensional manifold modeled on a Banach space, what does it mean for it to have a Riemannian metric? Does it necessarily mean that it is actually a Hilbert manifold? My ...
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1answer
280 views

Möbius transformation in the complex plane.

Assume that $U$ be a line in the complex plane. And assume a Möbius transformation $\phi $ sends $ U $ again to a line. How can I classify all such $\phi$? I want to write my ideas. But, I ...
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3answers
749 views

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 ...
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1answer
132 views

Is T($S^2 \times S^1$) trivial?

How would I find out if T($S^2 \times S^1$) is trivial or not? Using the hairy ball theorem I can show that T($S^2$) is not trivial, and it is straight forward to show that T($S^1$) is trivial. ...
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1answer
99 views

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 ...
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3answers
99 views

Embeddings (how to prove them exactly)

For which of the following sets is the statement: '$A$ can be embedded in $B$' true? I can try to decide this intuitively but don't know if I'm right, and surely don't know how to formally prove it. ...
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2answers
262 views

Dimension of de Rham Cohomology groups?

Is there a simple way to prove that the de Rham cohomology groups of a compact manifold $M$ have finite dimension as $\mathbb{R}$-vector spaces?
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1answer
488 views

A Cover of an Orientable Manifold is Orientable

The following question comes from Introduction to Smooth Manifolds by Lee: Suppose $\widetilde{M}$ smoothly covers $M$ where $M$ is orientable. Show that $\widetilde{M}$ is orientable. I think the ...
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2answers
81 views

What's the “easiest” closed 3-manifold with a nonabelian fundamental group?

I'm looking for some easy compact, oriented 3-manifolds without boundary that have a nonabelian fundamental group. It needn't be perfect. "Easy" means that it has an easy Heegard diagram, say, one ...
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2answers
90 views

Detecting compactness from the ring of smooth functions

Given a smooth manifold $M$, is there some ring-theoretic property (preferably not mentioning $M$) such that $C^{\infty}(M)$ has this property if and only if $M$ is compact?
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1answer
500 views

Complement of figure-8 knot

I am reading W. Thurston's famous "3-dimensional Geometry and Topology", but I am stuck at the point where it is said that gluing two tetrahedra in an appropriate way give you the complement of the ...
7
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1answer
437 views

Looking for an atlas with 1 chart

Can we provide the set $\{(x,y,z)\in\mathbb{R^3}|x^2+y^2=1\}$ with a 2-dimensional manifold structure involving only 1 chart? I can see it with 2 charts with cylindrical coordinates, but not with only ...
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1answer
2k views

The special orthogonal group is a manifold

How can we show that $SO(n)$ is an $n^2$-manifold. It would be tempting to say that $SO(n)$ is an open set of $\mathbb R^{n^2}$ but this is not the case since $SO(n)$ is given as the intersection of ...
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1answer
145 views

There is no immersion of the Möbius band in the plane.

There is no immersion of the Möbius band in the plane. I believe we have to work with the tangent bundle of the Möbius band, but I'm not getting no useful result.
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2answers
1k views

Why are Riemann surfaces algebraic curves?

I've never fully understood the connection between Riemann surfaces and algebraic varieties. I'm particularly interested in the case of the modular curve of level N--I know how the Riemann surface is ...
7
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2answers
192 views

Do we know this homogeneous space by another name?

Consider the homogeneous space $GL(3)/GL(2) = GL(3,\mathbb{R})/GL(2,\mathbb{R})$ where $GL(2)$ fixes the first coordinate axis (so can be identified with the subgroup of $2\times 2$ blocks sitting in ...
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1answer
832 views

Converse To Quotient Manifold Theorem [Exercise in Lee Smooth Manifolds]

I would like help with the following problem (chapter 9, #4) from Lee's Smooth Manifolds [its not homework, I'm reading it and I got stuck on this one] If a Lie group $G$ acts smoothly and freely on ...
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1answer
43 views

Grassmanians and boudaries of manifolds

Let $M$ be a smooth, compact manifold without boundary. I will say that $M$ is a boundary when there is a smooth, compact manifold with boundary $W$ such that $\partial W=M$. After some lectures I ...
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1answer
94 views

Is the Whitney embedding theorem tight for all $n$?

Whitney's embedding theorem states that any smooth $n$-manifold $M$ can be smoothly embedded into $\Bbb R^{2n}$. In dimension 1 this is tight: the circle cannot be embedded into $\Bbb R^1$. It is a ...
7
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1answer
94 views

Orientation of $X \times Y$

Suppose that $X$ is not orientable. How can I show that $X \times Y$ is never orientable, no matter what manifold $Y$ may be? I've tried supposing that $X \times Y$ is orientable, then using that ...
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2answers
142 views

Finding a subspace whose intersections with other subpaces are trivial.

On p.24 of the John M. Lee's Introduction to Smooth Manifolds (2nd ed.), he constructs the smooth structure of the Grassmannian. And when he tries to show Hausdorff condition, he says that for any 2 ...
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1answer
385 views

De Rham cohomology of $\mathbb{RP}^{n}$

Consider map from $S^{n}$ to $\mathbb{RP}^{n}$ $$\varphi:S^{n}\to\mathbb{RP}^{n}$$ which maps point $x\in S^{n}$ to corresponding direction in $\mathbb{R}^{n+1}$. This map induces map ...
7
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1answer
48 views

Angle form, 1-form, proof verification.

Check that the $1$-form $d\,\text{arg}$ in $\mathbb{R}^2 - \{0\}$ is just the form$${{-y}\over{x^2 + y^2}}\,dx + {{x}\over{x^2 + y^2}}\,dy.$$ My solution is as follows. Observe that we can ...
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1answer
83 views

Hurewicz map factors through bordism homology

I've read in multiple sources that the hurewicz map $h \colon \pi_n(X) \to H_n(X)$ factors through oriented bordism homology. I'm particularly interested in the injectivity of the map $h \colon ...
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2answers
106 views

What are some good sources to learn about real analytic manifolds?

Asking this question as someone with a graduate student level understanding of smooth differential/Riemannian geometry (May be a bit more that Riemannian Geometry by Do Carmo). I am trying to upgrade ...
7
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1answer
106 views

Show that $f\in A_{n-1}(V)$ or $f\in A_n(V)$ is decomposable (Tensors, or k-linear forms)

Show that $f\in A_{n-1}(V)$ or $f\in A_n(V)$ is decomposable. $f\in A_k(V)$ is decomposable if there exists a $a_1,...,a_k\in V^\wedge$ such that $f=a_1\wedge...\wedge a_k$ In this case "let ...
7
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1answer
137 views

Elliptic Regularity on Manifolds

Sorry if this has been asked before. Are there are books which talk about elliptic regularity on manifolds? Everything I've been able to find has been about $\mathbb{R}^n$. Can every question about ...
7
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1answer
156 views

What is the relationship between Grassmann Manifolds with different dimensions?

I'm an EE student and I'm just beginning to learn about the Grassmann Manifold. As is known that the Grassmann Manifold is a space treating each linear subspace with a specific dimension in the vector ...
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1answer
225 views

What's the intuition behind the tangent bundle?

Well, when we work with a smooth manifold $M$ we can associate with each point $p\in M$ a vector space $T_p M$ of all vectors at $p$ tangent to $M$: this is the space of linear functionals obeying ...
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1answer
389 views

Vector field and integral curve

Let M a riemaniann manifold, $V$ a vector field in M, and $\phi_t$ the respective flow Let $p\in M$, and $\gamma$ the integral curve of $V$. Show that for all $v\in T_pM$ ...
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0answers
149 views

Fundamental class of the connected sum of two closed orientable manifolds

I need to find a representation of $ [M \mathbin\sharp N] \in H_n(M \mathbin\sharp N) $ in terms of the fundamental classes $[M]$ and $[N]$. My idea is that $$ [M \mathbin\sharp N] = [M]+[N]$$ ...
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1answer
64 views

If M is a manifold of dimension $ n \neq0$ then M has no isolated points.

I am in doubt whether the following statement is true or false: "If M is a manifold of dimension $ n \neq0$ then M has no isolated points." The idea that made me find the true statement was as ...
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0answers
177 views

Showing L infinity norm bounded by L2 norm on a manifold

I have the following problem that I'm working on: Suppose $(M, g_{ij})$ is a compact Riemannian manifold. Assume $u$ is a smooth, nonnegative function which satisfies the differential inequality ...
7
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2answers
349 views

Seifert matrices and Arf invariant — Cinquefoil knot

I have computed the following Seifert matrix for the Cinquefoil knot: $$ S = \begin{pmatrix} 1 & -1 & -1 & -1 \\ 0 & 1 & 0 & 0 \\  0 & 1 & 1 & 0 \\ 0 ...
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6answers
2k views

What math is necessary to learn manifolds?

I want to learn about manifolds, but I'm only a senior in high school and obviously have a while to go. I'm in AP Calc BC. What should I study to eventually learn manifolds? Linear Algebra? What ...