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

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3
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2answers
87 views

Show $M_1\cap M_2$ submanifold iff $N_x(M_1)\cap N_x(M_2) = \{0\}$ and dimensions

Let $M_1,M_2 \subseteq \mathbb{R}^3$ two-dimendional submanifolds of $\mathbb{R}^3$ such that for every point $x\in M_1\cap M_2$ $$N_x(M_1)\cap N_x(M_2) = \{0\}.$$ Show $M_1\cap M_2$ is an ...
0
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0answers
44 views

Sections of the dual bundle of a smooth vector bundle

Let $M$ be a smooth manifold and $E,F,G$ smooth vector bundles over $M$. Denote the global sections by $\Gamma(.)$. In this question, it is proven that the canonical map ...
3
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0answers
31 views

A singular $n-$cube and a circumference defined the border than 2-cube

This is an exercise from "Calculus on Manifolds" by Michel Spivack (first edition, p.100): If $c$ is a singular $1$-cube in $\mathbb{R}^2-\{0\}$, with $c(0)=c(1)$, show that there is an integer ...
5
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2answers
390 views

Spin manifold and the second Stiefel-Whitney class

We know that: Spin structures will exist if and only if the second Stiefel-Whitney class $w_2(M)\in H^2(M,\mathbb Z/2)$ of $M$ vanishes. Can someone use simple words and logic to show why the ...
0
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1answer
34 views

Is there a local flow that is diffeomorphic at any time?

This question is regarding p.223-224 of Loring Tu's Introduction to Manifolds (Second edition). Without proof, the author previously assumed the following. Theorem. Let $M$ be a manifold and $X$ be ...
0
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0answers
51 views

Calculate the geodesic of Z = XY between two points

I have only learned about calculus and linear algebra, so I don't know about differential algebra. I got to know about the concept of "geodesic" recently. What I need to know is this: Suppose I ...
2
votes
1answer
77 views

Connection vs Curvature

Why is twice a connection usually referred to the curvature: $\overline{\nabla}\circ\nabla=F^\nabla$ Is there an axiomatic definition of curvature, e.g. it is module-linear operator etc?
0
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1answer
27 views

one problem on multivariable claculus

Suppose $\phi(\bar{x}(t))$ be a function which takes vectors (parameterized by $t$) as argument. Now take $c$ be a minimum point of the function $\phi$. consider a curve $\gamma(t)$ which passes ...
0
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1answer
28 views

Dual isogenies of complex tori in Birkenhake-Lange

Let $f: X\to Y$ be an isogeny of complex tori, of degree $n$. On page 13 of Complex Abelian Varieties, Birkenhake-Lange show that there is a dual isogeny $g: Y\to X$. Basically, they show ...
2
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0answers
32 views

Degree One Map induces Surjections on Homology

Is the following statement true: If $f:M\to N$ is a degree one map of compact closed manifolds, then $f$ induces surjections $f^*:H_q(M)\to H_q(N)$. I found this claimed on ...
1
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1answer
29 views

Examples of finding norm-minimizing surfaces and Thurston polytope

Let $Y$ be a (compact, oriented, connected) $3$-manifold. Thurston introduced a norm on $H_2(Y, \partial Y)$, which is defined as follow: any class $x \in H_2(Y, \partial Y; \mathbb{Z})$ is ...
0
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0answers
55 views

Stokes theorem on manifold without boundary

I'm struggling to understand why the integral should vanish: $\partial M=\varnothing:\quad\int_Md\omega$ For example: $0=\int_{B_1(0)}d(ydx)=\int_{B_1(0)}1dy\wedge dx=\mu(B_1(0))\neq 0\text{ ?}$
0
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2answers
86 views

Homotopy invariance of line integral on manifolds

Consider a 1-form: $\omega\in\Gamma(\mathrm{T}^*M)$ and two differentiable curves: $\gamma,\tilde{\gamma}:[a,b]\to M:\gamma(a)=\tilde{\gamma}(a),\gamma(b)=\tilde{\gamma}(b)$ together with a ...
2
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1answer
57 views

Spivak Calculus on Manifolds, problem 1-2

I am confused about the hint Spivak adds to problem 1-2 in his Calculus on Manifolds: When does equality hold in Theorem 1-1(3)? Hint: Re-examine the proof; the answer is not “when $x$ and $y$ ...
1
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1answer
32 views

Branched covering of a manifold [duplicate]

What would be the definition of a branched covering of a manifold? I am not familiar with branched coverings at all.
0
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1answer
55 views

Soft Question: Geometry of Rings

So.. I was thinking about something last night and literally have no idea where to start. Can the ring axioms be connected to topology? If so, how? Basically, the motivation for my question has to ...
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0answers
36 views

Manifolds and magnetic potential

Assume we have a particle in $\mathbb{R}^3$, which we will subject to different fields independently. It will have some potential energy $U\in \mathbb{R}$ defined as some constant minus its kinetic ...
2
votes
1answer
53 views

Is a non-compact Riemannian manifold a “measure space”?

One can define $L^p$ spaces for measure spaces with a given measure. Is a non-compact (i.e., it has a boundary) bounded Riemannian manifold a measure space? I am thinking of the manifold $(0,T) \times ...
1
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2answers
165 views

Definition of “a topological manifold with corners”.

How can we define a topological manifold with corners and its corners? Then, do we use "invariance of domain" to define corners, as we really need this theorem in order to define "boundaries of a ...
10
votes
1answer
94 views

Is the E8 manifold homeomorphic to a CW complex?

Is the E8 manifold homeomorphic to a CW complex? (I know that it is not triangulable) Edit: The E8 manifold is the unique compact (without boundary), simply connected topological 4-manifold, whose ...
1
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0answers
43 views

A basic question on intuition of second countability for definition of topological manifold

What is the intuition behind the requirement of second countability in definition of topological manifold? It seems that it is relevant to the $\mathbb{R}^n$ has second countable basis. Is there any ...
3
votes
1answer
42 views

Is the collection of atlases on a set $X$ a set?

Well, the title says it all. I need to know if i can view the collection of all atlases on a given set $X$ as a ordinary set. Is this possible ? All the atlases are only topological atlases, no ...
3
votes
1answer
79 views

Are two spaces obtained from homeomorphic spaces by removing a ball still homeomorphic?

I have a specific example in mind. Consider $S_1,S_2$ two surfaces. Remove two discs to obtain surfaces with boundary $S_1',S_2'.$ If $S_1 \cong S_2,$ does it necessarily follow that $S_1' \cong ...
10
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0answers
155 views

Soft question: How does basic differential geometry “fit together”?

I'm self-studying diff geom from Lee's Introduction to Smooth Manifolds. He warns the reader that there's a lot of machinery to construct, which is fine, and he explains things with wonderful clarity. ...
2
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0answers
39 views

Topologies of flag manifolds

I'm currently reading an article discussing flag manifolds and the action of $\mathrm{PSL}(n,\mathbb{C})$ on them. A flag (in my view at least) is a nested sequence $(y^1,\ldots,y^{n-1})$ of subspaces ...
2
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1answer
61 views

Euler characteristic for non-compact manifolds

How can one generalize the Euler characteristic to non-compact manifolds? Furthermore, is there a way to generalize the notion of an intersection number to non-compact manifolds, so that one could ...
1
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1answer
65 views

Which of principal curvature, Gaussian curvature and mean curvature is intrinsic ? Why?

Which of principal curvature, Gaussian curvature and mean curvature is intrinsic and why? Which book will have this explanation? I read a book (Remannian Geometry, Do Carmo, p129). It has a saying: ...
1
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1answer
136 views

Another differential topology lemma

Another lemma (1) Why can we assume $z=f(z)=0$ and that $U$ is convex? (the coordinate domains of the manifolds can be taken to be balls?) (2) Why is it enough to consider the special case of a ...
2
votes
1answer
59 views

Proving that something is a manifold

I'm a beginner at differential geometry and I'm having some trouble with the following problem: Let $M \subset \mathbb{R}^n$ be a $k$-dimensional smooth manifold (smoothly embedded in ...
1
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1answer
21 views

Cont. Function smooth iff composition with submanifold inclusion is smooth

I'm trying to proof the following: Let $X$ be a smooth manifold, $X_0$ an open subset of $X$, $i: X_0 \to X$ the canonical inclusion, $Y$ another smooth manifold and $f: Y\to X_0$ continuous, then ...
3
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1answer
46 views

Diffeomorphic connected hypersurfaces

Given a four dimensional Lorentzian manifold $\mathcal{M}$ (a manifold with a metric $g_{\mu\nu}$ in the tangent bundle with signature (-1, 1, 1, 1)), we define a global spatial foliation by a ...
1
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1answer
41 views

Linear Subspace of $\mathbb{R}^n$ is a Manifold

How does one prove that a that a linear subspace of $\mathbb{R}^n$ is a manifold? This question arises from Spivak's Calculus on Manifolds, Chapter 5, problem 5-5: Prove that a k-dimensional ...
1
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1answer
39 views

For which values of $a$ is this set a manifold?

Let $f:\mathbb{R}^3\to\mathbb{R}, f(x,y,z)=(x-y+z-1)^2$. For which values of $a$ is $\{(x,y,z)\in\mathbb{R}^3:f(x,y,z)=a\}$ a 2-manifold? Instead of $(x-y+z-1)^2=a$ seems a better idea to write ...
1
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1answer
34 views

Prove that inverse of $f$ defines a manifold

Let $f:\mathbb{R}^3\to\mathbb{R}$ be given by $f(x,y,z)=z^2$. Prove that $0$ is not a regular point but $f^{-1}(\{0\})$ is a manifold. I divided this in two parts: $(1)\; 0$ is not a regular ...
0
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1answer
167 views

A differential topology lemma

Consider the following lemma (1) How come he talks about degrees here, after all he doesn't assume $X$ to be oriented? (2) Why is $\bar{v}|\partial X$ homotopic to $g$? (NOTE: we consider them as ...
1
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1answer
71 views

Trying to prove M is a manifold

Let $M$ be the set of all points $(x, y, z) \in \mathbb{R^3}$ satisfying both of the equations $x^3 + y^3 + z^3 = 1$ and $x + y + z = 1$. Prove that M is a manifold, except perhaps near the points ...
2
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0answers
26 views

Charts on a Manifold

Let $f^{1},\cdots ,f^r,\Phi ^1,\cdots , \Phi ^{n-r}$ be functions of class $C^{(1)}$ on an open set $D$. suppose that $F=(f^1\mid S,\cdots ,f^r\mid S)$ is a coordinate system for $S$, that $S=\lbrace ...
0
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0answers
34 views

existance of loop with finitely many point of intersection

for every loop on compact orientable surface exists freely homotopic loop with finitely many points of intersection. I see that it have to be true, but I can't prove it. I know Thom's theorem, Sard's ...
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0answers
30 views

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 ...
1
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1answer
35 views

Integral Curves of a Vector Field

How do I find the integral curves of a vector field and what are they intuitively? eg. what are the integral curves of vector field $X=\frac{1}{x}\frac{\partial}{\partial ...
1
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1answer
64 views

The completion of $C_c^\infty(M)$ with respect to $\lVert \nabla u\rVert_{L^2(M)}$ on a compact Riemannian manifold

Let $M$ be a compact smooth Riemannian manifold (eg. a smooth hypersurface) and let $X$ be the space given as the completion of $C_c^\infty(M)$ with respect to the norm $\left(\int_{M} |\nabla ...
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0answers
28 views

Basic Manifold Question

Below is a paragraph from the appendix from Krantz's Several Complex Variables book. I have limited knowledge regarding manifolds and was hoping (very much) that someone would be willing to provide ...
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0answers
68 views

Why is surgery along a framed link well defined?

Let $L=L_1 \cup L_2 \cup \cdots \cup L_n $ be a framed link in $ S^3 $. I want to perform the surgery along $L$ to get a new manifold $M$. By definition, to perform this surgery, I must perform the ...
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0answers
89 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 ...
0
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1answer
32 views

Alexander duality formulation + Jordan-Brouwer separation

In Davis & Kirk LNAT p.71 there is written: (1) How does this imply the Alexander duality $\tilde{H}^k(A)\cong \tilde{H}_{n-k-1}(\mathbb{S}^n\!\setminus\!A)$? (2) Is it assumed that the ...
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3answers
58 views

What is a “control point”?

I'm trying to figure out a good definition of control point for use in wikipedia (see https://en.wikipedia.org/wiki/Control_point_(mathematics) ) There seems to be a bias towards ascribing a ...
0
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0answers
38 views

Can the second order frame bundle of a 2-d manifold be embeded inside the ordinary frame bundle of a 3-d manifold?

I would like to know whether the second order frame bundle of a 2-d manifold can be embedded inside the ordinary frame bundle of a 3-d manifold? Explanation Suppose we have a 3D smooth manifold ...
0
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1answer
49 views

When can you recover a connection from totally geodesic submanifolds?

Let $g_{ab}$ a Riemaniann ( Lorentzian ) metric in a $n-$dimensional manifold $N$ and let $M$ be a submanifold of $N$. In general, the Levi-Civitta connection induced by the induced metric in $M$ ...
2
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0answers
28 views

Connected sum of surfaces with boundary

The connected sum of closed surfaces (2-manifolds) is defined by removing a disk from each and gluing the exposed edges together. When defining the connected sum of surfaces with boundary, is the ...
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0answers
28 views

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 ...