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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408 views

Pullback distributes over wedge product

I'm looking to prove that the pullback of a smooth function distributes over wedge product, i.e. $$\varphi^*(\omega \wedge \eta) = \varphi^* \omega \wedge \varphi^* \eta. $$ Here the question is ...
2
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1answer
59 views

How do connection 1-form and Ehresmann version of connections relate to each other?

I am studying connections on abstract manifolds. So far, I have read several equivalent definitions but I can't establish the equivalence between them on my own. The first definition is the Ehresmann ...
2
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1answer
29 views

How to construct the horizontal bundle?

I am learning the concept of connection. I am confused by the construction of the horizontal bundle. My question is: For a fibre bundle $M\rightarrow B$, the vertical vector space $V$ can be easily ...
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35 views

derivative of flow

If I have a vector field $V$ on a manifold $M$ with flow $V_t$, and a curve $\gamma(s):\mathbb{R}\to M$, how do I compute $$\frac{d}{ds}\Bigg\vert_{s=0} V_t \gamma(s)?$$ I expect it to be a tangent ...
3
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1answer
930 views

Pulling back vector fields

I want to find conditions under which one can pull-back vector fields (if it is at all possible). Let $F:M \to N$ be a smooth surjective map between two $C^{\infty}$ manifolds of the same dimension. ...
2
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2answers
40 views

Smooth self maps of compact manifolds.

Suppose $M$ is a compact $n$ dimensional manifold. Does there exist an example of the following: A smooth map $f: M \rightarrow M$ such that there exists $x \in M$ where $\mbox{d}_x f$ has maximal ...
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1answer
32 views

Killing Field on a Riemannian Manifold

Do there exist a nontrivial Killing field on each riemannian manifold? A Killing field is a vector field whose flow acts on the manifold by isometry.
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0answers
34 views

What is an illustrative example of a Finsler manifold?

I've attempted to get a bead on Finsler manifolds before attending an upcoming seminar that involves them. I've done some reading in the literature and think I understand that they are similar to ...
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0answers
38 views

Orientability and volume form

I was wondering if there is an easy argument that manifolds that are orientable always have a volume form and vice versa? So I am not looking for a full proof of this, but rather a good argument how ...
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0answers
45 views

gluing manifolds along boundaries

I have a problem with the following task. Suppose $M_1, M_2$ are smooth manifolds with boundary. Let $f_1, f_2$ be diffeomorphisms from $B_1$ to $B_2$ ($B_i$ - the boundary of $M_i$), and suppose ...
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3answers
67 views

Can we bypass connection?

I am new to differential geometry. It is surprising to find that the linear connection is not a tensor, namely, not coordinate-independent. Can we bypass this ugly object? Only intrinsic quantities ...
1
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1answer
44 views

Counterexample about representation of parametrized curves.

In my book it says that in $\mathbb{R}^3$ there are parametrized curves which cannot be seen as the intersection of surfaces given by the expressions $F(x,y,z)=0,G(x,y,z)=0$. Is there in ...
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0answers
19 views

Charts in an oriented manifold with boundary

Let $M$ be an oriented manifold (with boundary) with $dim (M)\ge 2$. Show that there exists an atlas $\{(U_{\alpha},\phi_\alpha)\}_{\alpha\in I}$ for the chosen orientation such that ...
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1answer
28 views

Differential geometry: restriction of differentiable map to regular surface is differentiable

From Do Carmo: Let $S_1$, $S_2$ be regular surfaces. Suppose $S_1\subset V\subset \mathbb{R}^3$ and $\varphi:V\rightarrow \mathbb{R}^3$ is a differentiable map such that $\varphi(S_1)\subset S_2$. ...
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1answer
41 views

Value of the derivative of a function at a point depends only on the germ at that point

Suppose that f : I → R is a $C^∞$ function defined on an open subset I ⊆ R. How can I show that for $a \in I$ the value $f^n (s)$, n = 1, 2, 3, . . . of the derivative of $f$ of order n at s depends ...
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115 views

Advanced Differential Geometry Textbook

In algebraic topology there are two canonical "advanced" textbooks that go quite far beyond the usual graduate courses. They are Switzer Algebraic Topology: Homology and Homotopy and Whitehead ...
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0answers
53 views

About the diameter of a Riemannian manifold

My exact problem is a Riemannian manifold defined on $SU(2^n)$, where the metric is defined as follows: If $U(t)$ is a curve so that $U′(t)=−iH(t)U(t)$ (so just a unitary evolution of a quantum system ...
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1answer
28 views

Is such kind of manifold Riemannian? Deforming the metric on the unit square by a weight applied in one direction

If the metric is defined on a bounded subset of the x-y plane,let's say a closed square area $0\le x,y\le1 $, the metric is defined as $$\langle u,v\rangle =\langle (u_x,u_y),(v_x,v_y)\rangle =\langle ...
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0answers
44 views

Is there a generalization of the Quaternionic Hopf fibrations and its natural connection?

For the quaternionic Hopf fibrations, $S^3\rightarrow S^7\rightarrow S^4$, we have a natural BPST connection form. Do we have some generalization of it? For example, the 'natural' connection form on ...
2
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1answer
44 views

Non-orientable submanifolds

Let $M$ be a $n$-manifold and let $S \subset M$ be a non-orientable $n$-dimensional submanifold possibly with boundary. Under what conditions can I conclude that $M$ is also non-orientable? Is ...
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0answers
31 views

Integrate symplectic two-form

I am supposed to show the following (maybe falsely stated) theorem Let $\Phi$ be a symplectic group action $\Phi: G \times M \rightarrow M$ on a manifold $M$. Assume that the symplectic form $\omega$ ...
3
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0answers
85 views

Equivalent condition for non-orientability of a manifold

I've just came across this question, which gives us a great tool for showing that smooth manifold is non-orientable. Namely Thm. If $M$ is a smooth manifold and there are two charts ...
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0answers
26 views

Computing Gauss curvature using Meusnier theorem

I have troubles with finding Gauss curvature and mean curvature in a certain point of an oblique cylindrical surface. I know the way using the fundamental forms, but I am supposed to use the Meusnier ...
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4answers
659 views

Why do we need Hausdorff-ness in definition of topological manifold?

Suppose $M^n$ is a topological manifold, then $M^n$ locally looks like $\mathbb{R}^n$. $M^n$ is locally Hausdorff, since $\mathbb{R}^n$ is Hausdorff and Hausdorff-ness is a topological invariant. ...
2
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1answer
37 views

The Differential Geometry of a 2-D Surface

I'm currently self-studying the differential geometry of embedded surfaces. My question is, how am I to chose the appropriate coordinates and derive the covariant basis for the surface I'm interested ...
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38 views

Understanding the definition of Lie bracket

This is the definition I was given of Lie bracket: Let be $M$ a differentiable manifold and $v$ and $w$ two vector fields on $M$. The Poisson bracket $[v,w]$ between $v$ and $w$ is a vector field on ...
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1answer
40 views

Constant rank theorem: intuition?

Let $f: \mathbb R^n \to \mathbb R^m$ be smooth and let $x_0 \in \mathbb R^n$ be such that $\operatorname{rank}{(J_f(x_0))} = k $. Then there exists a neighboudhood of $x_0$ and diffeomorphisms $\phi, ...
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0answers
41 views

an isomorphism between the tangent space of a manifold to euclidean space

1) I was told in class many years ago that, the tangent space if the sphere $\mathbb{S}^2$at a point $p$, i.e. $T_p\mathbb{S}^2$ is isomorphic to $\mathbb{R}^2$. Could anyone give me a proof of ...
2
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0answers
23 views

Parallel transform of a vector by Lie derivative

I am new to differential geometry and I learn by myself. It seems that we need something extra called a connection to parallel transport a vector along a curve. But, suppose we have a vector field ...
1
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1answer
37 views

Generalizations of Inverse Function Theorem

A beautiful exercise in Guilleman and Pollack asks us to show the following generalization of the Inverse Function Theorem: Suppose $f: M \to N$ is a map of smooth manifolds, and $Z$ is a ...
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1answer
35 views

Vector field on $S^2 \setminus \mathsf{NP}$ looks like a magnetic dipole

The following is a question from Spivak's Differential Geometry text: Not really sure what he's going for here. Any ideas?
2
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1answer
71 views

flat manifold, curvature and the circle

A Riemannian manifold is said to be flat if the curvature is 0 everywhere. An example in dimension 1 is the circle. However, I cannot see how the curvature of the circle could be 0. See for instance ...
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1answer
123 views

Is anyone talking about “ball bundles” of metric spaces?

In differential geometry: Each smooth manifold $M$ is equipped with a tangent bundle $TM,$ which is a manifold equipped with a projection back to $M$ Given a smooth map $f : M \rightarrow N$ between ...
2
votes
1answer
90 views

Manifold is not orientable

Let $M$ be a manifold of dimension $n$ such that there exist two charts $(U_a,\phi_a)$ and $(U_b,\phi_b)$ such that $U_a,U_b$ are connected and $U_a\cap U_b\ne\emptyset$. Moreover the ...
4
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1answer
37 views

Definition of complex submanifold

For smooth manifolds, we can define an embedded submanifold to be either (1) a subset locally cut out by "slice" charts, or (2) a subset that is a manifold in the subspace topology and admits a smooth ...
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1answer
38 views

Looking for a biholomorphism [closed]

Is there a biholomorphism between $\mathbb{H}_2 \times S^1$ and a half-space of $\mathbb{R}^3$?
0
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1answer
30 views

Diameter of a Riemannian manifold on $SU(N)$ with almost negative curvature everywhere.

Are there any results (papers/books) on this problem? I am working on a finite dimensional Riemannian manifold which has a negative curvature almost everywhere. But I do not know if such kind of ...
2
votes
0answers
34 views

Why is the singular $p$-simplex called singular?

I am learning homology. I cannot see in which sense the singular $p$-simplex is singular. It seems quite regular. Any idea?
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0answers
38 views

Reference request: Pullbacks along submersions are submanifolds and the induced map is a submersion

I'm looking for an introductory book in differential topology in which there are proofs of the following facts: Let $X,X',Y$ be smooth manifolds, $X\rightarrow Y$ a smooth map and $X'\rightarrow Y$ ...
3
votes
1answer
186 views

Lie bracket; confusing proof from lecture

I am having some difficulties understanding this proof. Let $G$ be a closed matrixsubgroup of the general linear group. We have a right translation $Y(g):=dR_g(e) Y(e)$ on the Lie algebra $Y \in ...
2
votes
1answer
23 views

Can $S^4$ be the cotangent bundle of a manifold?

I am asking the question because in the classical mechanics book by Arnold, he states that there is a distinguished 1-form on $T^*V $. It seems that there is no such distinguished 1-form on a general ...
1
vote
1answer
140 views

Parallelizability of Lie groups

I have tried a lot to prove this well-known result. The basic idea behind the proof is clear to me. But I'm stuck at showing either of the following conditions: The map $\ G \rightarrow TG \ $ given ...
7
votes
1answer
121 views

Laplace-Beltrami on a Curve

Is there a way to write out Laplace-Beltrami operator explicitly for a sufficiently smooth plane curve given by implicit equation $\,s(x,y)=0\,$? If I knew knew the parametrization of the curve I ...
2
votes
1answer
31 views

vector field on $\mathbb{R}^n$ versus on manifold

I am looking for a counter example that why the $\mathbb{R}^n$ definition of vector field fail on a manifold. The following is a summary of what I learnt few years ago. Start with the idea of ...
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0answers
34 views

Relation between integral curves

Let $M$ be a manifold, $f:M\rightarrow\mathbb{R}$ a differentiable map and $X$ a vector field on $M$. I'm trying to find a relation between the integral curves of $X$ and $e^fX$. I am not quite sure ...
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0answers
44 views

What if there is $\downarrow$ or $\uparrow$ notation in the limit instead of $\rightarrow$?

I saw a different notation in a limit in the book Elementary Differential Geometry by A N Pressley : what do both of $\downarrow$ and $\uparrow$ mean?
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1answer
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How does an atlas give a notion of whether a function is differentiable or not?

Defintion. An atlas is a set of pairs $\{(O_i,ψ_i)\}$ such that $\cup_iO_i=M$ and $\phi:O_i \rightarrow \mathbb{R}^n$, and most importantly, for $O_i \cap O_j \neq \emptyset $, $\tau_{ij}: \phi_i(O_i ...
2
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2answers
40 views

Volume forms and volume of a smooth manifold

Choose a volume form $\omega$ on $M$, oriented manifold. For every $F\in C^{\infty}_c(M)$, we define $$ \int_M F:=\int_M F\omega $$ where in the right hand term $M$ is taken wit positive orientation ...
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0answers
46 views

I have no idea what “smooth structure” is

I know what a manifold is: it's a topological space such that for every point there is an open set that looks like $\mathbb{R}^n$. But I do not know what a smooth manifold is, because I have no clue ...
6
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4answers
111 views

Show that Lie bracket is in Lie algebra?

Let $so(3)$ be the Lie Algebra of $SO(3)$ and $R\in SO(3); \Omega_1,\Omega_2 \in so(3)$ and $\Omega_n = \frac{d}{dt}R_n(t)$ at the point $t=0$. So $\Omega_n$ is the tangent vector of the curve ...