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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Is there a natural Riemannian structure on the total space of a vector bundle?

Suppose $B$ is a Riemannian manifold and $\pi: E \to B$ is a smooth vector bundle equipped with a metric. Is there a natural Riemannian metric on $E$, i.e. a bundle metric on $TE\to E$? It seems ...
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48 views

Why $\dfrac{\partial \sigma}{\partial u}=\dfrac{\partial \sigma}{\partial \bar{u}}$?

According to Elementary Differential Geometry by A N Pressley: (${\bf \sigma}_u=:\dfrac{\partial \sigma}{\partial u}$). The above text several times assuming that $\dfrac{\partial ...
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1answer
34 views

If two objects have the same gaussian curvature, are they the same up to isometries?

I was reading about Gauss Egregium Theorem but I'm not sure if I understand it well. Intuitively, what does it mean? It is true that if two objects have the same Gaussian curvature, then they are the ...
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1answer
14 views

Show that the outward unit normal is smooth vector field

Exercise 2.1.8 of Guillemin and Pollack asks us to prove that if $X^m \subset \mathbb{R}^n$ is an embedded submanifold with boundary, then the outward unit normal to $\partial X$ is a smooth function ...
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1answer
41 views

Symplectic form and wedge sum

The wedge sum of $k$ symplectic 2-forms is given by ( if $\omega = \sum_i e_i^* \wedge f_i^*$) $$ \omega^k = k! \sum_{1\le i_1 <...<i_k \le n} (e_{i_1}^* \wedge f_{i_1}^*) \wedge ... \wedge ...
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1answer
29 views

Is the canonical bundle topologically trivial?

Suppose $X$ is a $n$-complex dimensional complex manifold, we can form its canonical bundle $K_{X,\mathrm{hol}}=\bigwedge^n\Omega_{X,\mathbf{C}}$. Usually this bundle is not holomorphically trivial. ...
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36 views

A diffeomorphism which maps geodesics to geodesics preserves the connection?

Let $(M,\nabla^M),(N,\nabla^N)$ be two smooth manifolds with given (affine) connections on their (tangent bundles). We say a diffeomorphism ,$\phi:(M,\nabla^M)\rightarrow(N,\nabla^N)$ is an ...
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1answer
46 views

Why derivation is a directional derivative?

Suppose $M$ is a smooth manifold, and $X\in T_pM$. Why for every derivation in $p\in M$ exist tangent vector $X\in T_pM$, witch satisfies $L_p(f) = X\cdot f$ for every smooth $f\colon M\to\mathbb{R}$? ...
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1answer
29 views

The complex structure of a complex torus

A complex torus $X$ of complex dimension $1$ is just a quotient of $\mathbb{C}$ by a lattice $\Lambda=\mathbb{Z}\oplus \tau\mathbb{Z}$ where $\tau=a+ib$ is a complex number with $b>0$. The complex ...
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2answers
64 views

Definition of Cech-De Rham complex. Can't understand its definition!!!!

Let $M$ be a manifold and let $\mathcal{U}:=\{U_\alpha\}_{\alpha\in I}$ be an open covering, $I$ be a totally ordered set. For every $p$ and for every $\alpha_0<\dots<\alpha_p$, $$ ...
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1answer
27 views

Prove that $\int_V (\mathbf{x} - \mathbf{x}_c) dx dy dz= 0$, $x_c$ is the centroid of the volume $V$

This is what I came up with and I am not sure if it is correct, and I would like to know if there is another, maybe purely geometrical, way of obtaining the equation. A centroid will be the center ...
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12 views

Proof of co-area formula in terms of level set function

How to prove the following theorem, Let for each $t \in [0,T]$, $\phi( t,\cdot) : {\bar \Omega} \rightarrow R$ be Lipschitz continuous and assume that for each $r \in ( \textrm{inf}_{\Omega} \phi, ...
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1answer
50 views

Proof of Brouwer fixed point theorem using Stokes's theorem

$\omega$ is the volume form on the boundary B -ball $f\colon B \to \partial B$ $$ 0 < \int_{\partial B}\omega = \int_{\partial B} f^*(\omega) = \int_{B} df^*(\omega) = \int_B f^*(d\omega) = ...
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17 views

Diffeomorphism and Orientable double cover

Suppose that the orientable double cover of two homeomorphic surfaces are diffeomorphic, is it true that these surfaces are diffeomorphic?
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14 views

One-sided surfaces and the second variation area formula.

I know how to find the second variation area formula for a two-sided minimal embedded surface in a 3-manifold and the condition for such a surface to be stable. But, what about one-sided surfaces? ...
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hard work of mathematician [on hold]

how many hours did all greatest mathematician work ?
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26 views

Proving a given set is a submanifold

Let $S \subseteq \mathbb R^n$. I have been faced with showing that $S$ is a submanifold and I have some ideas but I want to get the complete picture. (Main) Question 1: What methods are there to ...
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31 views

Is there some connection between the killing form and a killing vector field?

I'm studying a paper on globally symmetric spaces, and the methods involved use at some points Killing vector fields. Since globally symmetric spaces can be well-understood by methods of Lie algebras, ...
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35 views

Prove the differential of $f$ at $p$ is a well defined linear map

Let $W$ be a subset of $\mathbb{R^n}$ be open and $f:W \rightarrow \mathbb{R^m}$ be smooth. The differential of $f$ at $p \in W$is a linear map: $$df_p: \mathbb{R^n} \rightarrow \mathbb{R^m}$$ ...
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1answer
34 views

A proof for the Mobius Strip parametrization

According to Elementary Differential Geometry by A N Pressley, a parameterization for Mobius strip is : And according to Wiki another parameterization is $x(u,v)= \left(1+\frac{v}{2} \cos ...
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1answer
40 views

Lie bracket of two vectors $X,Y$ perpendicular to $Z$ is perpendicular to $Z$

Where $Z$ is a Killing vector field (is this even necessary?) In case more assumptions are necessary, I have: $[Z,X] = [Z,Y] = g(Z,X) = g(Z,Y) = 0$ I want to prove $g([X,Y],Z) = 0$ I am trying to ...
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1answer
36 views

Locally flat manifold from Frobenius, differential forms approach

It is a known fact that a Manifold $M$ is locally flat if and only if the Riemann curvature tensor (Levi-Civita connection) vanishes. To show the if part is easy, but I am trying to follow the details ...
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1answer
39 views

Complex-valued differential forms.

Let $X$ be smooth (real) manifold and let $T^{*}(X)_{\mathbb{C}}$ denote the complexification of the cotangent bundle. We define the complex valued differential r-forms on $X$ to be the smooth ...
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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 ...
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27 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 ...
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0answers
29 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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35 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 ...
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1answer
28 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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34 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 ...
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2answers
39 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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Locus of points on a curve for constant segment lengths squared sum $ OM^2 + MP^2 $ [on hold]

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 ...
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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)} $ ...
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32 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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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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36 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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37 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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18 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
43 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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1answer
40 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
65 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 ...
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1answer
49 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 ...
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1answer
40 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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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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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 ...
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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 ...
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1answer
26 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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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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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 ...
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1answer
38 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, ...