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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On the Spivak's proof of the theorem 3-11 (calculus on manifolds)

In second paragraph of the case 1 within the proof: What is $U$ s.t $A\subset U$ and satisfies in the following my second question? $\psi_i$ is defined on $U_i$ and its support is not compact. Now, ...
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Metric tensor for n-sphere in ambient coordinates

Let $S^n$ be the unit n-sphere embedded in $\mathbb{R}^{n+1}$: $$ S^n = \{ a \in \mathbb{R}^{n+1} \mid a \cdot a = 1 \} $$ What is the induced metric tensor for the sphere, in $\mathbb{R}^{n+1}$ ...
4
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27 views

Ricci curvature along Killing vector field

If $V$ is a Killing vector field, I need to prove that $$V^{m}\nabla_{m}R = 0$$ where $R$ is the Ricci scalar $R = g^{mn}R_{mn}$. I´m having some trouble with this, I already showed that ...
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2answers
95 views

Closed form on any submanifold closed?

Let $M$ be a manifold and $\omega$ a closed differential form, so i.e. $d \omega =0.$ If I now consider a submanifold $N$ of $M$. Does this mean that $\omega$ is still closed on $N$? This statement ...
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43 views

Why is this not an inconsistency in elementary Lie theory?

I made an observation last week, and it has bothered me ever since. Recall the formulae ...
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19 views

Sending unitary group to reals, second differential at critical points.

Let$$X = U(n) = \{B \in \text{GL}_n(\mathbb{C}): BB^* = I\}$$be the group of $n \times n$ unitary matrices over $\mathbb{C}$, viewed as a real manifold. Fix a diagonal $n \times n$ matrix $A$ with ...
3
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1answer
42 views

Lie Derivative Equals to Lie Bracket

I am reading the book Introduction to Smooth Manifold written by M.Lee. I am confusing with the concept of Lie derivative. We have $\mathcal{L}_XY=[X,Y]$. However we have ...
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1answer
25 views

Linearization of divergence of a vector field?

Let's $X$ is a fixed smooth vector field on semi-Riemannian manifold $(M,g)$. For a symmetric 2-tensor field $s$, and for sufficiently small values of $t$, $\tilde{g}=g+ts$ is a semi_Riemannian metric ...
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1answer
47 views

Computing Rham Cohomology

Suppose that we have a $C^{\infty}$ manifold $X$ with and atlas $\mathcal{A}=$($U_{\alpha},\varphi_{\alpha}$) such that for every two intersecting open sets $U,V \in \mathcal{A}$ the intersection is ...
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2answers
33 views

Uniqueness of tangent plane

Let $\Sigma$ be a smooth surface defined as a surface admitting a parametrisation $\boldsymbol{r}:D\subset\mathbb{R}^2\to\mathbb{R}^3$ such that $\boldsymbol{r}$ is of class $C^1(\mathring{D})$ (and ...
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Something about a restriction of a smooth function over a surface. [on hold]

Let $F$ be a smooth function in $\mathbb{R}^3$ and let $f$ be the restriction of $F$ to a surface $\Sigma$. Show that $$\mathcal{Lf}=\sum_{i=1}^{2}(\bar{D}^2F)(e_i, e_i)-\frac{1}{2}<x, ...
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37 views

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 ...
3
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1answer
56 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 ...
2
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1answer
41 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
15 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
45 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
34 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. ...
3
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39 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
47 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
31 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
65 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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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
51 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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0answers
18 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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15 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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42 views

hard work of mathematician [on hold]

how many hours did all greatest mathematician work ?
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1answer
36 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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0answers
41 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 ...
3
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1answer
43 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 ...
2
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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 ...
2
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0answers
57 views

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
29 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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0answers
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 ...
1
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0answers
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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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)} $ ...
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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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37 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 ...
1
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0answers
40 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
42 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 ...