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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Variation of geodesic and Jacobi field

I was reading on Jacobi field of a geodesic, and noticed that given a geodesic $\gamma$ it is defined using the term of variation or family of geodesics $\gamma_s$ but never mentioned how to create ...
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1answer
27 views

Computing the first fundamental form

Let $X$ be a smooth surface in $\mathbb{R}^{3}$. I want to compute the first fundamental form of $X$. Assume that $X$ has 2 different local parameterizations $r_1$, $r_2$ (i.e. for $r_1$ there is a ...
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38 views

Poincaré Lemma problems and computing contractions in an economical way

Let $x=(A,B,C,D)$ be coordinates on $\mathbb{R}^4$. $\displaystyle \beta = \frac{(AdB-BdA)\wedge(dC \wedge dD)+(dA \wedge dB)\wedge(CdD-DdC)}{(A^2+B^2+C^2+D^2)^2}$ I would like to compute ...
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32 views

Show that $d\beta=0 \iff p=n/2$

Let $\beta$ be the $(n-1)$-form on $\mathbb{R}^n \setminus \{0\}$ given by $\displaystyle \beta = \sum_{i=1}^{n}(-1)^{i-1}\frac{x^i dx^1 \wedge dx^2 \wedge \dots \wedge \hat{dx^i} \wedge \dots ...
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36 views

Does this surface exist

Does anybody of you know if there is a surface with first fundamental form $(g_{ij}) = \operatorname{diag}(1, \cos^2(u))$ and second fundamental form $(h_{ij}) = \operatorname{diag}(1, \sin^2(u))$? ...
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32 views

Why are there more complex than smooth structures?

I've read that given a topological manifold, there are only finitely many smooth structures on it (except for dimension 4) but many more (even uncountably many) complex structures. But doesnt this ...
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1answer
24 views

Curvature line parametrization

I have a question about the curvature line parametrization. We said that for a given surface $f: U \rightarrow \mathbb{R}^3$ we find a local curvature line parametrization such that both the first ...
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99 views
+50

Use of Poincare Lemma in solving $\nabla \times \textbf{A}(\textbf{r})=\frac{\textbf{r}}{r^3}$

Let $U = \mathbb{R}^3 \setminus \{(0,0,z) \}$ (ie $\mathbb{R}^3$ with the $z$-axis removed ) and consider $\beta$ on $U$ given by $\displaystyle \beta = \frac{x dy \wedge dz + y dz ...
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50 views

determine the principal curvatures of the surface defined as the tube around a space curve using the Frenet Serret frame.

Consider a regular unit speed curve $\alpha: (a,b) \to \Bbb R^3$. Then define the surface $S$ via the parametrization $x:(a,b)\times (-\pi,\pi)\to \Bbb R^3$ where $$x(u,v) = \alpha(t) + r(N(t)\cos ...
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1answer
49 views

Manifold with special cohomology group

I am trying to find if there is a orientable compact manifold $M$ of dimension 10 with the 5th cohomology group of De Rham $H^5_{DR}(M)\cong \mathbb R$. But, I can find such an example or prove that ...
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25 views

definition of differential does not depend on the choice of the curve, differential is a linear map

Let $\varphi: M \to N$ be a differentiable map. How do I show that the definition of the differential $d\varphi_p: T_pM \to T_{\varphi(p)}N$ of $\varphi$ at $p$ does not depend on the choice of the ...
3
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1answer
29 views

Every immersion is locally an embedding?

Let $\varphi: M \to N$ be an immersion and let $p$ be a point in $M$. How would I go about showing that there exists a neighborhood $V \subset M$ of $p$ such that the restriction $\varphi|V$ of ...
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1answer
61 views

Identity concerning Lie derivative of $k$-form $\omega$

Let $X$ and $Y$ be vector fields on $\mathbb{R}^n$. Show that for $\omega$, a $k$-form on $\mathbb{R}^n$, $(L_XL_Y-L_YL_X)\omega=L_{[X,Y]}\omega $. I try using Cartan's magic formula and get that ...
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51 views
+50

Identity of the pushforward of a vector field using a Jacobi bracket.

Let $Z(u,v)$ be the vector field $Z(u,v)=(u^2+u,v^2+v)$, let $\Gamma_t$ denote its flow. I have shown that $[X,Z]=Z-X$. Show that $(\Gamma_t)_*X=e^{-t}X-(e^{-t}-1)Z$. Could someone please show me ...
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25 views

Green's operator, differential forms

In "Foundations of Differential Manifolds and Lie Groups" by Frank Warner on page 225 there is defined Green's operator: $G: E^p(M) \rightarrow (H^p)^{\perp}$ by setting $G(\alpha)$ to equal the ...
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22 views

no point where assumptions of Inverse Function Theorem hold?

Let $\varphi: \mathbb{R}^3 \to \mathbb{R}$, $\psi: \mathbb{R}^3 \to \mathbb{R}$ are continuously differentiable. Define $f: \mathbb{R}^3 \to \mathbb{R}^3$ by$$f({\bf x}) = (\varphi({\bf x}), \psi({\bf ...
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27 views

Geodesic loops in Riemann homogeneous spaces

Let M be a Riemannian homogeneous space, i.e. the isometry group acts transitively. Prove: any geodesic loop (with possible angle at the starting point) is a closed geodesic (smooth at the starting ...
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1answer
60 views

Sectional curvature of product metric?

If $M$ and $N$ are Riemannian manifolds, can we relate the sectional curvature of the product Riemannian manifold $M \times N$ to those of $M$ and $N$? If both $M$ and $N$ have non-negative (or ...
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119 views

Differential forms, number of zeros on disk

Let $f$ be a holomorphic function on $U \subset \mathbb{C}$ and $f'(z) \neq 0$, $z \in U$. Then how do I show that all zeros of $f$ are simple and positive, and that if I have a disk $D \subset U$ ...
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1answer
51 views

Set of smooth maps between manifolds is a smooth manifold

If $M$ and $N$ are smooth manifolds of dimension $m$ and $n$, respectively. Is set of smooth maps between them a smooth manifold? With which smooth structure?
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1answer
30 views

Definition of covariant derivative

Let $E \to M$ be a vector bundle (with fibres $V$) over a smooth manifold $M$. Define the covariant derivative $\nabla$ as a map $$\nabla : C^\infty(M,E) \to C^\infty(M,E \otimes T^*M),$$ where ...
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29 views

Finding the Lie derivative of a 2-form exercise

Let $\beta=-x dx \wedge dy + ydy \wedge dz$. The vector field is $X=(y,0,z)$.Find the Lie derivative. I try that $\begin{align}L_X \beta &=L_X (-x dx \wedge dy + ydy \wedge dz) = -x L_X(dx ...
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86 views

Cancellation in topological product

I was wondering whether $M\times \mathbb{R}$ is homeomorphic to $N\times \mathbb{R}$ implies $M$ is homeomorphic to $N$, where let us say $M,N$ are smooth manifolds. (They are certainly homotopy ...
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1answer
18 views

Identity about composition of the push forward of diffeomorphisms

I am able to do part a) and I believe it should be used in solving part b). I think that for part b) we should that $F: \mathbb{R}^n \rightarrow \mathbb{R}^n$, $G: \mathbb{R}^n \rightarrow ...
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18 views

Delaunay surfaces - plane as surface of revolution

According to Wikipedia (and other sources) "Delaunay proved that the only surfaces of revolution with constant mean curvature were the surfaces obtained by rotating the roulettes of the conics. These ...
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1answer
55 views

Relying two points by an almost-geodesic omitting a singular set a.e.

I failed to give an appropriate title to the question, so any suggestion for a better title is welcome: Here's the question: I would like to prove the following result: Given $\varepsilon>0$, a ...
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0answers
19 views

Degree of a map $\phi: R^{n} \rightarrow S^{n}$

I've read a few papers in which they state that the winding number of a mapping $\phi: R^{3} \rightarrow S^{3}$ can be written as the integral $$\int_{\mathcal{R}^3} \epsilon_{ijk}\epsilon^{abc} ...
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1answer
46 views

Eembedding of product $\mathbb{S}^2\times\mathbb{S}^3$ into $\mathbb{R}^6$

It is easy to see that $\mathbb{S}^n$ can be embedded in $\mathbb{R}^{n+1}$ and therefore $\mathbb{S}^2\times\mathbb{S}^3$ can be embedded in $\mathbb{R}^7$. The question is how to prove that ...
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50 views

Riemann manifolds in relation to other classes of differentiable manifolds

I am trying to get an overview over the different categories of manifolds. In particular i have the following chain of inclusions: Riemann surfaces $\subset$ complex manifolds $\subset$ orientable ...
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1answer
38 views

Laplace-de Rham operator on $\mathbb{R}^n$

Let $\mathbb{R}^n$ have the standard orientation with volume element $dV = dx_1 \wedge...\wedge dx_n > 0$. Show that $\Delta = - \sum_j \partial^2 / dx_j^2$ on 0-forms on $\mathbb{R}^n$. Where ...
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1answer
81 views

differential forms, cylindrical coordinates, geometric interpretation [closed]

Consider a differential $1$-form $\beta$ which in cylindrical coordinates $(r, \theta, z)$ has the form $$\beta = f(r)\,dz + g(r)\,d\theta,$$where $g'(0) = 0$. Find a condition when $\beta \wedge ...
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139 views

Curvature of De Sitter's space: where does the sign comes?

Consider $\Bbb L^3 = (\Bbb R^3, {\rm d}s^2)$, where: $${\rm d}s^2 = {\rm d}x^2 + {\rm d}y^2 - {\rm d}z^2.$$ We have both the hyperbolic space: $$\Bbb H^2(-1) = \{(x,y,z) \in \Bbb L^3 \mid ...
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2answers
52 views

How to define a vector field passing through a given point.

Given a manifold $M$ we can consider its tangent bundle $TM$. Fix $m \in M$ and $v \in TM$. Is it always possible to define a vector field $F$ such that $F(p)=v$?
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1answer
48 views

Can Spivak's 5-volume series on differential geometry be effective without exercises?

I was scouring the internet for information about these books and I learned that the latter 4 volumes have no exercises. Would I be able to attain mastery with no exercises?
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Simply-connected Lorentzian manifold and event horizon

Can a simply connected Lorentzian manifold admit an event horizon? Or does the event horizon makes it non-simply connected?
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1answer
61 views

Induced Lie group action on a tangent bundle $TG\times TM\to TM$ and an example concerning Adjoint action

Suppose I have a Lie group action $$ G\times M\to M, (g,m)\mapsto g\cdot m. $$ which is transitive on $M$, then the tangent functor $T$ induces a corresponding map: $$ TG\times TM\to TM, (\delta ...
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2answers
122 views

geometric interpretation of $\vec v \cdot \operatorname{curl} \vec v = 0 $

There is a family of surfaces orthogonal to the vector field $\vec v \in \mathbb R^3$ iff $\vec v \cdot \operatorname{curl} \vec v = 0 $. Now the necessity part is trivial, but the proof of ...
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19 views

Oval surface with $ K \ge 1 $ which is unit sphere

Let $ S$ oval surface with $ K \ge 1$ . If there is exist an open unit sphere interior of $ S$, then $S$ is unit sphere...Can anyone give me an idea of the solution...thanks in advance...
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1answer
12 views

Out of plane cross section evolution of surfaces based on local geometry information

With this question I would like to kindly ask for feedback or general pointers to even remotely related works in regards to a challenge I face. Given a smooth surface $S$ $:\mathbb{R}^2\rightarrow ...
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1answer
9 views

a surface tangent to a plane along a curve

I can't visualize how a surface is tangent to a plane along a curve. Tangent plane of a surface at a specific point intersect the surface at only one point, doesn't it ? Can someone help me to ...
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2answers
39 views

Reference request: Introduction to Applied Differential Geometry for Physicists and Engineers

I'm looking for a book on differential geometry or differential topology that is comprehensive and reads at the level of someone with engineering background (i.e. Boyce's ODE, Stewart's Calculus, ...
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1answer
48 views

“Rigid” Riemannian metrics

What do we mean when we say that a Riemannian metric $g$ is rigid? For example, the Eguchi-Hanson metric is rigid as an Einstein metric. Any help is appreciated!
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34 views

cohomology homomorphism induced by classifying map

Let $\xi=(E,p,B)$ be an $n$-dimensional vector bundle. Let $f: B\to G_n(\mathbb{R}^\infty)$ be the classifying map. Let $f^*: ...
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1answer
38 views

What is the difference between intrinsic and extrinsic manifold?

I'm asking this question because a course change on differential geometry at my university has updated the wording from extrinsic manifold to intrinsic manifold. This got me wonder as to what the ...
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19 views

Define the concept of the Shape Operator and Fundamental Forms

I am confused about the relationship between three concepts: shape operator, first fundamental form, second fundamental form. I would like someone to provide me with a basic definition of these terms ...
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1answer
41 views

$d(\beta \wedge d\beta)=0$ if $k$ is even.

Let $\beta$ be a $k$-form. Show that $d(\beta \wedge d\beta)=0$ if $k$ is even. I get that $d(\beta \wedge d\beta)=d\beta \wedge d \beta + (-1)^k\beta \wedge d^2\beta=d\beta \wedge d \beta$. Why ...
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2answers
23 views

Measure of the intersection of a ball and a compact subset

Do you know a large class of compact subset $K$ of $\mathbb{R}^d$ such that for each such compact $K$, there exists a $r>0$ with $\inf_{x \in K} \lambda^d(B_r(x) \cap K) > 0$, where $\lambda^d$ ...
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1answer
40 views

Fundamental forms of a composite function

So I have to work out the three fundamental forms, the principal and Gaussian curvatures for this surface element... I have to use the function: $f(u_1,...,u_n)$ = $(u_1,...,u_n, F(u))$ where F is a ...
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30 views

cohomology of finite dimensional grassmannians

What is the cohomology algebra of finite dimensional grassmannian $$ H^*(G_k(\mathbb{R}^N);\mathbb{Z}_2)? $$ $$ H^*(G_k(\mathbb{C}^N);\mathbb{Z}_p)? $$ $$ H^*(G_k(\mathbb{H}^N);\mathbb{Z}_p)? $$ I ...