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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What theorems or frameworks explain why the roots of well-behaved functions $h : \mathbb{R} \leftarrow \mathbb{R}^2$ seem to be made up of “pieces”?

First, some terminology: given functions $g,f:Y \leftarrow X$, the equalizer of $g$ and $f$ is defined to be the set of all solutions $x \in X$ to the equation $g(x)=f(x)$ in $Y$. Okay. The following ...
3
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0answers
30 views

Zero Gaussian curvature and restriction estimates

Let $$ \left(\int_M|\hat{f}|^qd\mu(\xi)\right)^{1/q}\leq c||f||_{L^p(\mathbb{R}^n)} $$ be a restriction estimate for a hypersurface $M\subset\mathbb{R}^n,~1<p<\infty$ and $\mu$ a ...
2
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30 views

Geodesics without a metric

By definition, a geodesic is a mapping $\gamma: I = (0, 1) \rightarrow M$ such that $\nabla_{\dot{\gamma}(t)} \dot{\gamma}(t) = 0$. Here we only need the connection. So, we do not need a metric to ...
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36 views

The second fundamental form of the sphere

I am trying to understand how one computes the second fundamental form of the sphere. Looking at the example on page 10. http://www.math.miami.edu/~galloway/dgnotes/chpt5.pdf Here I understood ...
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 ...
2
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0answers
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 ...
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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67 views

Locus of points on a curve for constant segment lengths squared sum $ OM^2 + MP^2 $ [closed]

EDIT : This edit supersedes the post and edits before it as it is simplified and freshly done once again. After sometime they would be deleted if ok. Anyway: Two points M and P in a plane (Origin ...
1
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1answer
69 views

Use of exclamation point

I'm quite puzzled by the use of an exclamation point in this paper. The authors introduce the following linear constraints to a quadratic program: $ \sum_k a^{(l)}_k b_j (\mathbf{x}_k) = r_j^{(l)} $ ...
2
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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 ...
1
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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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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
42 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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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
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 ...
2
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1answer
52 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 ...
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3answers
66 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 ...
3
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1answer
51 views

Specifying an arbitrary point on a manifold

It is known that any arbitrary point x on the sphere $\mathbb{S}^2$ can be parametrised by the spherical coordinates $$\bf{x}=r(\theta,\phi)=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta),\quad ...
0
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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 ...
5
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0answers
111 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
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
votes
1answer
43 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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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 ...
2
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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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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 $\omega$ ...
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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 ...
1
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1answer
39 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
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 ...
2
votes
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 ...
3
votes
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77 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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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
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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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 ...
2
votes
1answer
68 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 ...
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
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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
37 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 $X'\rightarrow Y$ ...
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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
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 ...
0
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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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vote
1answer
26 views

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 ...
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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 ...
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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 ...
2
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1answer
86 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 ...
2
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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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1answer
24 views

Representing a vector field locally

A couple of questions occured to me when reading about the Poincare-Hopf theorem. Any pointers or comments would be appreciated! Let $M$ be a closed oriented Riemannian manifold and $V$ a vector ...
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0answers
28 views

Curve as a linear combination of Frenet frame variables

How to write a curve $\alpha(s)$ as a linear combination of $\alpha'(s),n(s),b(s)$ where these are the tangent to the curve, the normal vector and the binormal vector. Where torsion is nonzero and ...