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Questions tagged [differential-geometry]

Differential geometry is the application of differential calculus in the setting of smooth manifolds (curves, surfaces and higher dimensional examples). Modern differential geometry focuses "geometric structures" on such manifolds, such as bundles and connections; for questions not concerning such ...

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Showing that a locus is a sub-manifold

I'm self-studying differential geometry using Frankel's ``The Geometry of Physics". The first problem (1.1(1)) is about determining whether or not the locus $$x^2+y^2-z^2 = c $$ is a submanifold in $\...
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Lipschitz normally embedded submanifold

Given $(X,d_i)$ a smooth compact connected submanifold with intrinsic distance $d_i$ embedded in $R^d$. Is it true that: $d_i(x,x')\leq K\Vert x-x' \Vert $ for some $K>0$? If that is the case, how ...
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3answers
37 views

Understanding Lie algebra of matrix Lie group

In my lecture, we gave a very sloppy (physics people ...) proof of the fact that the Lie algebra $\mathfrak{g}$ of a matrix Lie group $G$ is a subspace of $\text{Mat}_n(\mathbb{F})$. I am not ...
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1answer
39 views

Embedding a compact manifold in $\mathbb{R}^N$

I have an attempt at solving the following problem. This is not so much a question asking for a solution in general, but more on how to complete my own. Let $M^n$ be a compact smooth manifold. Show ...
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12 views

Directed curvature of a curve

So I have this following exercise Consider the curve given by the graph of the sine function $t \rightarrow (t,\sin(t))$. Determine the directed curvature at each point of this curve. Supposing that ...
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1answer
21 views

Submanifold of real projective space

Would you like to tell me how to prove that $\{[x_{0}:x_{1}:x_{2}]: x_{0}x_{1} + x_{2}^{2}\} \subset \mathbb{R}\mathbb{P}^{2}$ is a submanifold (of $\mathbb{R}\mathbb{P}^{2}$)?
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Which 3-manifolds can be cubulated?

I am trying to get a picture of what is currently known about cubulability of 3-manifolds, though cannot seem to find a good overview. I am personally most interested in compact 3-manifolds with ...
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3answers
34 views

Extending a volume form on $S^k$ to $\mathbb R^{k+1}$ such that the exterior derivative is zero.

As we know that $S^k$ is a properly embedded submanifold of $R^{k+1}$. Let $w$ be a volume for of $S^k$(a non vanishing top form). Then $dw = 0$ on $S^k$. Is it possible to extend $w$ to $R^{k+1}$ ...
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30 views

Divergence theorem over non-smooth manifolds

The divergence theorem on smooth surfaces is well understood and changes the integrand from div(X) to dot(X,n) where n is the normal of the boundary of the surface. Suppose my surface is peicewise ...
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2answers
52 views

Tangent space of preimage is the preimage of the tangent space

Let $M$ and $N$ be smooth manifolds with $S\subseteq N$ a submanifold, and assume a map $f:M\to N$ is smooth and transverse to $S$. Prove that $T_p(f^{-1}(S)) = (df_p)^{-1}(T_{f(p)}S)$ for some $p\in ...
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62 views

Differential of Hopf's map

Let $$h : \mathbb{C^2} \rightarrow \mathbb{C \times R} $$ $$h(z_1, z_2) = (2z_1z_2^*, |z_1|^2-|z_2|^2)$$ How do you find the differential of $h$ and show it is onto/surjective? I know that I can ...
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31 views

Spherical Triangle: Law of Sines with Clairaut's theorem

The spherical law of sines states that On the sphere $\mathbb{S}^2$ we consider a triangle, i.e. three points connected by geodesics. We denote the side lengths with $a, b, c < \pi$ and the ...
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2answers
25 views

Smooth bijection and tangent spaces

Let $f:M\to N$ be a smooth bijection between manifolds with same dimension. Do we necessarily have $$df_p(T_pM)=T_{f(p)}N.$$ I think it is probably not true. But I can't give a counterexample...
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1answer
25 views

Proving that the image of an injective, proper immersion is a manifold

I am trying to get through the proof of the statement "if $f: M \to N$ is injective, proper and an immersion, then $f:M \to f(M)$ is a diffeomorphism onto a submanifold". The proof I'm reading says ...
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6 views

Multi-Variable CoArea formula for Total Variation

The total variation of a scalar function over a 2-manifold can be defined using the coarea formula. If I now use a N-d function, and I want to integrate the frobenius norm of its gradient over the 2-...
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1answer
46 views

The sphere with three ends?

The preceding image take form Matthias Weber's Classical Minimal Surfaces in Euclidean Space by Examples notes is called the sphere with three ends. But what does it have to do with a sphere and why ...
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1answer
27 views

Almost volume preserving charts for a riemannian manifold

Let $M$ be riemannian manifold. Is it true that for all $p\in M$ and $\varepsilon>0$ there is a chart $\phi:U\rightarrow\mathbb{R^n}$ arround $p$ such that for all nonempty open $V\subseteq U$ $\...
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1answer
36 views

Why are map projections of the Earth not charts?

The Earth is a classic example of a 2D manifold. Looks Euclidean to us, but is most definitely curved. I am self teaching some differential geometry and I don't quite understand the difference ...
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1answer
28 views

What is the difference between a chart and a tangent space?

To my lay-person mind, a chart is a one-to-one function that maps an area on a manifold to a euclidean space of equal dimension. Then I understand a tangent space to be the space of vectors that are ...
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1answer
44 views

Determinant of the second fundamental form in Gauss's curvature

At this point on a Prof. Norman J Wildberger's presentation on Gauss's curvature and the the Theorema Egregium the curvature of a manifold $S$ at a point $p$ is written down as the determinant of the ...
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1answer
34 views

How does this example from Spivak that $H_c^n(\mathbb R^n) \ne 0$?

I am not sure how this integral that is being calculated using Stoke's theorem shows that the $n$th de Rham cohomology group with compact supports of $\mathbb R^n$ is not trivial. How does the fact ...
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1answer
85 views

How does a differential form looks on a matrix manifold?

I want to know how does a differential form looks in a matrix manifold. For example, given that the special linear group $$SL(n,R)$$ of all matrices with determinant 1 is a manifold, how looks a 1-...
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28 views

Christoffel symbols of the second kind transformation law

We want to show that the Christoffel symbols of the second kind transform like a connection. the Christoffel symbols of the second kind are given by: $$\begin{Bmatrix}a \\ bc\end{Bmatrix} = \frac{1}{...
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1answer
47 views

Is it trivial that the Levi-Civita connection can be pulled back to a isometric manifold?

Let $M, N$ be Riemannian Manifolds and $\phi: M \to N$ a isometric diffeomorphism. We know that we have unique Levi-Civita Connections $\nabla^M, \nabla^N$ on $M, N$ respectively. One can check that $\...
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1answer
14 views

Local Automorphisms of Cartan Geometries are determined by values at a point

Let $M$ be a manifold and $(P\to E\overset\pi\to M, \omega: TE\to\mathfrak g)$ a Cartan geometry on $M$. I have seen the following statement: Let $f:U\to V$ be a local automorphism where $U,V$ are ...
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22 views

Non periodic vector field on $\mathbb{R}^2$ [on hold]

Give an example of non periodic vector field on $\mathbb{R}^2 $. I found that gradient systems cannot have periodic orbits. so I picked this vector field: $X(x,y) = x+y$. 1) Is this a valid vector ...
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2answers
37 views

Confusion about quotient of the Lie group $\mathbb{S}^1$

I have read that given a Lie group $G$ and a closed subgroup $H$ then $G/H$ is a smooth manifold. I cannot explain though the following example: take as $G = \mathbb{S}^1$ and as $H =\{\pm 1\}$, $H$ ...
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1answer
36 views

Definition of Differential operator

Definition 2.2, page 19 Let $M$ be a smooth manifold and $E_i \rightarrow M$ be two smooth vector bundles. A PDO $P:\Gamma (M,E_0) \rightarrow \Gamma(M,E_1)$ of order $k$ is a a linear map which ...
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1answer
37 views

Interpretation of the Lie algebra of a Matrix Lie group

I'm looking for an intuitive explanation of the meaning of the Lie algebra for a matrix Lie group from a differential geometry perspective. Right now, the procedure I've been following is using the ...
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1answer
26 views

prove that a map is an embedding

let $f : \mathbb{R}^2 \rightarrow \mathbb{R}^{+*}$ be a smooth function. Prove that $g:\mathbb{S}^1 \rightarrow \mathbb{R}^2: u \rightarrow g(u) = f(u)u$ is an embedding. I found that $h: \mathbb{R}...
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1answer
22 views

basis of tangent space of a submanifold defined as a graph

Let $f : \mathbb{R}^2 \rightarrow \mathbb{R}$ be a smooth function. Let $G:=\{(x, y, f(x, y)) : x,y \in \mathbb{R}^2\}$ be its graph. Find a basis for $T_pG$ for a $p(x,y,z) \in G$. What I did: I ...
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1answer
14 views

smooth curve that is tangent to a 1-form kernel in every point

Let $α = dz - ydx \in Ω^1 (\mathbb{R}^3)$. Prove that $\forall p,q \in \mathbb{R}^3,\ \exists \gamma: [0,1] \rightarrow \mathbb{R}^3$ smooth, such that $γ(0)=p, γ(1) =q$ and $\gamma$ is tangent to $...
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1answer
28 views

find a vector field in $\mathbb{R^3}$ with specific properties

Let $α = dz - ydx \in Ω^1 (\mathbb{R}^3)$ Find a vector field Z in $\mathbb{R^3}$ such that $α(Z) = 1$ and $dα(Z,.) = 0$ What I did: I computed $d\alpha = dx \wedge dy $. Then let $Z=a\partial_x + ...
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1answer
37 views

Question on what constitutes surfaces in R3

Our lecturer gave us the following definition of a surface in $\mathbb{R}^3$: $\Gamma$ Is a surface in $\mathbb{R}^3$ if for all $y\in \Gamma$ there exists a coordinate patch $\sigma : D\subset\mathbb{...
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The idea behind Delzant construction of a toric manifold from a convex polytope

I am trying to understand how to visualize a symplectic toric manifold from its moment polytope, following chapter 29.4 in "Lectures on Symplectic Geometry" by Ana Cannas da Silva: https://people.math....
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Sard's theorem for orientation preserving diffeomorphism of the circle

thanks in advance for helping me. First I'll introduce some definitions: (1) Suppose that $f : \mathbb{S}^{1} \rightarrow \mathbb{S}^{1} = \mathbb{R} / \mathbb{Z}$ is an orientation preserving ...
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0answers
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Sufficient and necessary conditions for a change of coordinates to be locally invertible

Given a 2D coordinate change $(x, y) \mapsto (u, v)$, what are the sufficient and necessary conditions for the map to be invertible on every local neighborhood? For instance, the map, $$ u = x^3 \quad,...
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1answer
38 views

Existence of a Tubular neighborhood of a hypersurface

Suppose $H$ is a co-dimension 1 embedded submanifold of $M$. Let $X$ be a vector field on $M$ such that $\forall x \in H$, $T_xM = T_xH \oplus X_x$. Now I want to show that there exixts an open set $U$...
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Inverse problem : Finding a vector bundle (resp. with connection), given its characteristic class (resp. differential character)

Given a rank $n$vector bundle $\alpha :E \to M$, and an element $u \in H^k(BG, \mathbb{Z})$, $G=GL(n,\mathbb{R})$we can define its characteristic class $u(\alpha) \in H^k(M, \mathbb{Z})$ as $f_\alpha^*...
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Calculate deRham Cohomology un R^3

I want to calculate the Cohomology groups of $B_r-B_s$ $(r>s)$ where $B_r$, $B_s$ are solid balls on $\mathbb{R}^3$ and the boundary of $B_s$ is empty, I tried use Mayer Vietoris and the $U$,$V$ ...
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22 views

How smooth is a foliation by analytic curves?

Perhaps I am looking for guidance or a reference to solve the following problem: Take the interior of a triangle in $\mathbb R ^2$ and call it $U$. $$\\$$ Take a foliation of $U$ by one dimensional ...
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64 views

Continuity of the Euler characteristic with respect to the Hausdorff metric

Hadwiger's theorem of integral geometry states that all continuous valuations which are invariant under rigid motions are expressible in terms of the intrinsic volumes. The continuity property means ...
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Higher order covariant derivative notation

The covariant derivative of a tensor can be noted equivalently as $$\nabla_b X^a \qquad \mathrm{and} \qquad X^a_{\; ;b}$$ If we want to express a higher order covariant derivative $$\nabla_c \...
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28 views

Derivative Of Infimum - Embedded Submanifold Of $\mathbb{R}^n$

This is a follow-up question to: Derivative Of A Function Defined In Terms Of An Infimum Let $A$ be a compact, (smooth) embedded submanifold of $\mathbb{R}^n$ and let $\Phi : \mathbb{R}^n \...
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0answers
21 views

Partial differential operator is sum of order $0$ operators and composition of vector fields

Let $P:\Gamma(\Bbb R^n, E_0) \rightarrow \Gamma(\Bbb R^n, E_1)$ be a differential operator where $E_0$, $E_1$ are trivial vector bundles, with the standard bundle metric over $\Bbb R^n$. In page 28 ...
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1answer
24 views

Nontriviality of the Hopf Fibration

A simple question how to understand why even though locally $S^3$ is homeomorphic to $S^2\times S^1$, how do you see that globally this is not true?
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Zero constant mean curvature in Minkowski space versus in Euclidean space

There's a famous result in $\mathbb{R}^3$ which goes as: there are no compact minimal surfaces in $\mathbb{R}^3$.So the mean curvature cannot be zero in compact surfaces in $\mathbb{R}^3$. Now what's ...
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1answer
33 views

Is it true that any Riemannian metric on cylinder can be deformed to product metric?

Is it true that any Riemannian metric on cylinder $\Bbb S^1\times \Bbb R$ can be deformed to standard product metric? Are there some standard references about such classifications?
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Any closed hypersurface has at least one point where all curvatures are positive

Any closed hypersurface has at least one point where all curvatures are positive. How to prove this statement? I found that statement at the paper " Volume-preserving flow by powers of the mth mean ...
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No compact surfaces with constant causal character in $\mathbb{L}^3$ [on hold]

The title is all there is to it: I want to know why there are no compact surfaces with constant causal character in $\mathbb{L}^3$. If possible, some indications in the computing of the causal ...