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

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1answer
29 views

First and second fundamental form with rotational surfaces (check)

I'm working out some examples for surfaces in differential geometry. I was working out simple rotational surface, but I think I've done something wrong. Let $\gamma\left(t\right)$ a curve ...
3
votes
1answer
45 views

Calabi-Yau $3$-fold given as elliptically fibered manifold over $\mathbb{C}P^1 \times \mathbb{C}P^1$

Consider a Calabi-Yau three-fold given as an elliptically fibered manifold over $\mathbb{C}P^1 \times \mathbb{C}P^1$$$y^2 = x^3 + f(z_1, z_2)x + g(z_1, z_2),$$where $z_1$, $z_2$ represent the two $\...
0
votes
0answers
19 views

Orientation of boundary of manifold

I'm studying an example concerning orientation in manifolds: Let $U=\mathbb{R}^2\backslash\{(0,0\}$ and $A=\{(x,y):0<r^2\le x^2+y^2\le R^2\}$ be a circle ring. Then the boundary is $$ \partial A=\...
0
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2answers
31 views

How do you find the metric tensor for a given manifold?

Is there some general way to derive the metric tensor for a given manifold M? For example, how was the metric for the surface of a sphere $$ds^2=d\theta^2+\sin^2\theta \, d\phi^2$$ first derived?
1
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1answer
83 views
+50

Prove that $g$ is a submanifold: $g (t,u,v) = (t^2,u^2,v^2,\sqrt{2}uv, \sqrt{2}tv,\sqrt{2}tu)$

We consider $g : (t,u,v)\in \mathbb{R}^3 \mapsto (t^2,u^2,v^2,\sqrt{2}uv, \sqrt{2}tv,\sqrt{2}tu)\in\mathbb{R}^6$. I have to prove that $g(\mathbb{S}^2)$ is a submanifold of $\mathbb{R}^6$. $dg_{(t,u,...
1
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1answer
75 views

How to prove that there are only two kinds of 1-dim manifolds without boundary

I just know a conclusion that all 1-dim manifolds without boundary is homomorphism to $S^1$ or $\mathbb{R}$ , but I don't know how to prove it . Why is so ?
0
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3answers
66 views

How to find a parametrization for a torus?

I need to compute the surface area of the torus $$T^2=\{(x,y,z)\subseteq\mathbb R^3 \left(\sqrt {x^2+y^2}- R\right)^2+z^2=r^2\}$$ where $0<r<R$. I know I need to compute the metric tensor and ...
2
votes
1answer
21 views

$f:M\to N$ smooth manifold map. $F(x)=(x,f(x))\in M\times N$. For each $X\in \mathfrak{X}(M)$ there's a F-related $Y\in \mathfrak{X}(M\times N)$.

Let $M$ and $N$ be to manifolds and $f:M\to N$ smooth map. Define $F:M\to M\times N$ by $F(x)=(x,f(x))$. Show that for each $X\in \mathfrak{X}(M)$ there's a F-related $Y\in \mathfrak{X}(M\times N)$. ...
0
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1answer
90 views

Connection on Submanifold using given connection on ambient manifold

This result features in Hicks' Notes on Differential Geometry book.The theorem states that given $C^\infty$ fields X and Y on a submanifold M, we have $$\bar D_X Y=D_X Y+V(X,Y)$$ where $\bar D$ is a ...
2
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1answer
19 views

Let $X_i, Y_i$ be vector fields on the manifolds M and N. $X_i\oplus Y_j$ on $M\times N$. $[X_1\oplus Y_1,X_2\oplus Y_2]=[X_1,Y_1]\oplus [X_2,Y_2]$

Let $M$ and $N$ be two differentiable manifolds and $X_1,X_2$ be two vector fields on $M$ and $Y_1, Y_2$ on $N$. Using the fact that $T_p(M)\oplus T_q(N)$ is naturally isomorphic to $T_{(p,q)}(M\times ...
2
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0answers
52 views

rho invariant of manifold

If $G$ is a finite group, then the rational oriented cobordism group $\Omega_{2k-1}^{Stop}(BG)\otimes{\mathbb Q}=0$, so if $N^{2k-1}$ is an orientable odd-dimensional Top manifold with fundamental ...
1
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1answer
16 views

Jacobian determinant of a function $f$ if it has a differentiable inverse.

This question is from the book "Differential Topology First Steps" by Andrew Wallace. The question is as follows: Let $f$ be a differentiable map of an open set in Euclidean n-space into n-space and ...
1
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1answer
22 views

If $M^n$ is a submanifold of $\mathbb{R}^{n+k}$ such there are L.I. vector fields $v_1,\ldots,v_k$ such that $v_i \perp T_pM$ then $M$ is orientable

If $M^n$ is a submanifold of $\mathbb{R}^{n+k}$ such there are L.I. vector fields $v_1,\ldots,v_k$ such that $v_i \perp T_pM$ for every $p \in M$ then $M$ is orientable. My attempt is: Once $M^n$ ...
2
votes
1answer
33 views

Viewing complex projective space as a Grassmannian manifold

Complex projective space $\mathbb{CP}^n$ carries the structure of a complex manifold of dimension $n$, hence has the underlying structure of a real manifold of dimension $2n$. It is the set of complex ...
25
votes
2answers
278 views

Origins of Differential Geometry and the Notion of Manifold

The title can potentially lend itself to a very broad discussion, so I'll try to narrow this post down to a few specific questions. I've been studying differential geometry and manifold theory a ...
0
votes
0answers
35 views

Distributions on submanifolds

I am beginner in differential geometry. I stuck with the concept of distributions(like invariant, anti invariant, slant) on submanifolds. Can you explain what are distributions on submanifolds? If ...
0
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0answers
17 views

Principal curvatures from parametrisation

Let $M^2$ be an immersed surface of the standard sphere $S^3$ with unit normal vector field $\eta : M \to \mathbb{R}^4$ (tangent to $S^3$). Given a point $p \in M$ and a parametrisation $\varphi : U \...
4
votes
1answer
36 views

Poincaré–Bendixson theorem on the torus

I was reading the paper A Generalization of a Poincaré-Bendixson Theorem to Closed Two-Dimensional Manifolds by Arthur J. Schwartz which proves the following theorem: THEOREM. Let $M$ be a ...
0
votes
1answer
22 views

Definition of manifold which are subset of euclidean space

According to Guillemin and Pollack "X(which is a subset of R^n)is a k-dimensional manifold if it is locally diffeomorphic to Rk , meaning that each point x possesses a neighborhood V in X which ...
0
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0answers
27 views

sectional curvature of hyperbolic space

For a pair of $(X,Y)$ of linearly independent vectors in $T_pM$, $p\in M$, the sectional curvature is defined as $$K_p(X,Y)= - \frac{<R(X,Y)X,Y>}{|X|^2 |Y|^2 - <X,Y>^2}$$ The problem I'm ...
3
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1answer
41 views

Construct smooth mapping $f: B^{n + 1} \to S^n$ with two singularities at which $f$ has degree $+/- 1$.

I'm currently working through a paper by Pjotr Hajlasz who wants to show that For smooth manifolds $M,N$, if $\pi_{[p]}(N) \neq 0$ and $1 \leq p < n = \dim M$, then the smooth mappings $C^\...
0
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0answers
44 views

addition of two differential forms with different degrees

Does it make sense to add two differential forms with different degrees like $dx+dx\bigwedge dy$? If yes, what's the arguments of it? I ask this because in text book, the vector space, $\Omega^*(M)$, ...
-1
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0answers
11 views

Transformation of vector from vector field coordinates

Consider an n-dimensional manifold with coordinates $x_1, x_2, \dots, x_n.$ Suppose we have a vector field defined on this manifold $V : \bar v = \bar v(\bar x).$ Let us perform a homogeneous ...
0
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1answer
19 views

Construct a diffeomorphism $\psi: B_1 \to \epsilon\text{-neighborhood of } K$, where $K$ is a subset of a smooth manifold.

I'm currently working through a paper by Brezis on the topology of Sobolev spaces. Right now I am having trouble understanding the following note made by Brezis. Let $M$ be a compact and smooth ...
3
votes
1answer
44 views

Do I have the right idea about affine connections?

On a smooth manifold $M$, a vector field is a smooth map $X : M \to TM$, where $TM$ is the tangent bundle of $M$. If $\chi(M)$ denotes the space of vector fields on $M$, an affine connection $\nabla$ ...
1
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1answer
49 views

Vector fields (on a manifold) and terminology

I read in several books (Do Carmo, Riemannian Geometry or John M. Lee, Smooth manifolds) that a vector field $X$ on a smooth manifold $M$ is a mapping which associates to each point $p \in M$ a ...
-1
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0answers
31 views

A exercise of Riemannian geometry . [closed]

In picture below,I don't know how to start the second question . It is obvious that the isometry of $R^3$ keep the dimension , so there exist such isometry. But seemly, it is too simple . Besides, ...
0
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0answers
21 views

How to prove that two compact 1-dim Riemannian manifolds with same length must be isometric?

How to prove that two compact 1-dim Riemannian manifolds with same length must be isometric ? I know the compact 1-dim manifold must be homeomorphism to $S^1$ , but how to do a specific isometric ?
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0answers
20 views

Torsion and curvature of a linear connection

Could you help me to solve the following problem ? Let $M$ a parallelizable manifold of dimension $n$, {$E_1$,...,$E_n$} a global frame of $M$. Let $X$,$Y$ a vector fields on $M$ with $Y= \sum_{i=1}^...
9
votes
1answer
166 views

When can we recover a manifold when we attach a $2n$-cell to $S^n$?

I have a question related to this one. In my answer I was going to try and say something about the possible manifolds that might arise in this way, i.e. as mapping cones of elements of $\pi_{2n-1}(S^...
8
votes
2answers
88 views

Homology and cohomology of 7-manifold

I have the following problem: Let $M$ be a connected closed $7$-manifold such that $H_1(M,\mathbb{Z}) = 0$, $H_2(M,\mathbb{Z}) = \mathbb{Z}$, $H_3(M,\mathbb{Z}) = \mathbb{Z}/2$. Compute $H_i(M,\...
0
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0answers
16 views

Computationally Determine Dynamics on a Center Manifold

I have a four dimensional system of nonlinear differential equations $\dot{x} = f(x)$ with a single parameter $\alpha$ and Jacobian matrix $J$ with eigenvalues $\lambda_1 = 0, Re(\lambda_2) < 0,Re(\...
1
vote
1answer
32 views

Where does $f(r,\varphi,\theta)=(r\sin{\theta}\cos{\varphi},r\sin{\theta}\sin{\varphi},r\cos{\theta})$ have a locally differentiable inversion?

$$f(r,\varphi,\theta)=(r\sin{\theta}\cos{\varphi},r\sin{\theta}\sin{\varphi},r\cos{\theta})$$ $$f:(0,\infty)\times\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}^3$$ How can I find out on which ...
0
votes
1answer
30 views

What is $u^{-1}TN$ with $u: M\rightarrow N$ be a smooth map

As picture below, $u\in C^\infty(M,N)$, $(M,g)$ and $(N,h)$ are two smooth Riemannian manifold. I don't know what mean the $\frac{\partial }{\partial y^1} \circ u$ , it is $\frac{\partial u}{\partial ...
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3answers
102 views

Why doesn't coordinate difference between two points correspond to distance between two points?

I know that in Euclidean geometry, where the manifold is "flat" (such that it is isomorphic to an open subset of $\mathbb{R}^{n}$), $M\cong\mathbb{R}^{n}$, one can use Cartesian coordinates, $\phi (p)\...
1
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1answer
48 views

A Riemannian manifold with constant sectional curvature is Einstein. [closed]

A Riemannian manifold with constant sectional curvature is Einstein. Why? It's true the inverse?
1
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1answer
36 views

Topology given by atlas is paracompact

I'm currently reading Jeffrey M. Lee Manifolds and Differential Geometry book. I don't understand a part in the proof of Proposition 1.32. (iii). Proposition 1.32. says: Let $M$ be a set with a $...
1
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2answers
59 views

Why a Riemannian manifold minus one point is not complete? [closed]

Could you give me a proof that a Riemannian manifold minus one point is ever complete? Thanks!!
1
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1answer
28 views

Integration of differential form on ellipsoidal surface with singularity in origin

As picture below ,I want to compute the (2) , because there is a singularity in $\{0\}$ and $\omega$ is closed . So ,I have $$ \int_M\omega=\int _{\partial B_1(0)} \omega $$ I think there is a ...
3
votes
1answer
431 views

Partial derivative of a function on manifold

Bishop and Goldberg define ("Tensor analysis on manifolds") the partial derivative of a smooth function on a manifold $M$ in the following way: $\partial_i f= \frac{\partial f}{\partial{x^i}}=\frac{\...
0
votes
1answer
53 views

What is a singular space?

A book I am reading on orbifolds uses the word singular space but doesn't say what it means. The book is Orbifolds and Stringy Topology by ALR the quote is "Orbifolds are singular spaces that are ...
0
votes
1answer
30 views

Differentiable functions between manifolds are continuous

Let $f:M \to N$ be differentiable function between manifolds. I want to show that $f$ is continuous. First, that $f$ is continuous should mean (correct me if I'm wrong!) that for every point $a\in N$ ...
0
votes
1answer
42 views

Prove is compact

I'm trying to solve this problem and I don't know how to start Let $M$ be a connected time-oriented Lorentz manifold of dimension $n$. Let $J^+(K)=\{q\in M: \text{there is a $p\in K$ with $p\leq q$}\}...
1
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1answer
51 views

Uniqueness of connected sum

Connected sum is defined as Wiki .But I think the result of connected sum is not unique. For example ,make connected sum on $S^2$ with itself . Then , the result can be $T^2$ or Klein bottle. Is it ...
2
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0answers
21 views

Introduction to morse theory with applications to optimization

I am wondering if there are any easy-to-read introduction materials on morse theory (especially with applications to nonconvex optimization) for people with non-math background.
1
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1answer
20 views

Optimization: Via manifolds point of view of Lagrange multipliers method

My basis on differential manifolds calculus and differential geometry being very superficial, I'm trying to understand this section on WP's article. I'm not being able to realize why most of the ...
3
votes
3answers
711 views

$M \times N$ orientable if and only if $M, N$ orientable

For two manifolds $M$ and $N$ I'm trying to prove that $M \times N$ is orientable if and only if $M$ and $N$ are orientable. My attempt so far: $\impliedby)$ Assume $M, N$ are orientable. Then $\...
-1
votes
2answers
39 views

Open subset of $\mathbb {CP}^n$? [closed]

As picture below, how to show the $U_i$ is open subset of $\mathbb {CP}^n$ ?
1
vote
1answer
42 views

Is $\mathbb R^2\backslash \{0\}$ a manifold?

Is $\mathbb R\backslash \{0\}$ a manifold ? Is $\mathbb R^2\backslash \{0\}$ a manifold ? I would say yes, but in the doubt, I prefer to ask.
0
votes
1answer
26 views

Is the subgroup of homotopically trivial isometries a closed subgroup of the isometry group?

Let $(M,g)$ be a connected Riemannian manifold. Then according to the Steenrod-Myers-Theorem, the isometry group $\text{Isom}(M,g)$ of $(M,g)$ is a compact lie group with the compact-open topology. ...