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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Let $\alpha(s)$ be a unit speed curve in $R^2$. Show $\kappa=|\frac{d\theta}{ds}|$

I'm lost on solving the following problem. Let $\alpha(s)$ be a unit speed curve in $R^2$. Show $\kappa=|\frac{d\theta}{ds}|$, where $\theta$ is the angle between the positive $x$-axis and the ...
4
votes
2answers
69 views

Killing vector field along a geodesic

I was trying to show that a Killing vector field satisfies the Jacobi Equation for a geodesic, just by assuming that \begin{equation} \nabla_\mu X_\nu + \nabla_\nu X_\mu=0 \end{equation} Indeed, if I ...
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1answer
33 views

A diffeomorphism whose tangent map preserves dot products is an isometry.

I'm having trouble solving the following problem. If $F:\mathbb{R^3} \to \mathbb{R^3}$ is a diffeomorphism such that $F\ast$(the tangent map of $F$) preserves dot products, show that $F$ is an ...
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28 views

An orthogonal transformation with determinant 1 rotate $\mathbb{R^3}$ around an axis.

I'm studying differential geometry and need confirmation on the solution of the following problem. A rotation is an orthogonal transformation $C$ such that det $C=+1$. Prove that $C$ does, in fact, ...
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1answer
33 views

How to show that two vector fields commute?

Could anyone help me with how to start to solve the following problem? From this problem as well as this, I have: Fix $\varepsilon \in (0, 1)$ and choose a smooth function $h$ on $[0,\infty)$ such ...
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0answers
13 views

Associated bundle - valued 1 forms.

Let $\pi:P\rightarrow M$ be a principal bundle with structure group $G$. Let $\mathfrak{g}$ be the Lie algebra of $G$. Fact: (as can be found in lecture notes here) Functions ...
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34 views

Vector fields that are smooth on the open unit ball $B$, are smooth on $\mathbb{R}^n$ if they are zero outside $B$

Could anyone help a little with the following problem? It is a continuation of this problem, but I will restate the things that are needed: Fix $\varepsilon \in (0, 1)$ and choose a smooth ...
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1answer
60 views

$\psi ( \frac {\pi}{2}, \frac {\pi}{6})$ and calculating problems? [closed]

I ran into a problem, $u=\psi (x,t)$ be a solution of partial deferential equation with following condition on boundary, how we reach the value of $\psi ( \frac {\pi}{2}, \frac {\pi}{6})$? ...
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votes
3answers
41 views

Derivative of a function in $\mathbb{R}^n$

Let $f:\mathbb{R}^m\to\mathbb{R}^m$ be a differential function. Let $Df(x)$ be the derivative of f at $x\in\mathbb{R}^m$. Which of the following is/are correct? $Df(0)(u)=0 \forall u\in\mathbb{R}^m$ ...
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0answers
24 views

Existence of a differentiable function given a unit gradient field

I'm trying to prove that "Given a unit vector field $V$, it can always be uniquely determined a differentiable function $f$ that satisfies $\nabla f = V$." To provide you more information, the unit ...
2
votes
1answer
24 views

Basic properties of smooth curves

Suppose $\Gamma$ is simple smooth closed curve parametrized by $\gamma:[0,1]\to\Gamma.$ Let $$\gamma(t)-\gamma(s)=(t-s)F(t,s)\,\,\,\,t,s\in[0,1].$$ Can we conclude that $|F(t,s)|>0$ for all ...
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32 views

What is some elementary differential geometry textbook that is self contained and are intermediate level?

What are some intermediate differential geometry textbook that are more advanced than pressley's, Barrett's, Christian's and krezig's books and are self contained but below the level spivak's vol ...
4
votes
3answers
47 views

is every totally geodesic submanifold the set of fixed points of some isometries?

It is well known that the set of fixed points of an isometry $\phi:(M,g)\rightarrow (M,g)$ is a totally geodesic embedded submanifold. (e.g here ). I ask whether the converse is true, i.e is every ...
2
votes
1answer
73 views

Proving a certain function is injective

I have found the following exercise on an exam for Geometry three dating to a past year. Let $F(u,v)=((2-v\sin\frac{u}{2})\sin u,(2-v\sin\frac{u}{2})\cos u,v\cos\frac{u}{2})$, with ...
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0answers
21 views

Proof that any unit-speed-reparametrization of a curve preserves orientation and is an inverse of an arc length function based at some $t_0$.

I am not able to prove the following two facts about a unit speed reparametrization of a curve. Let $\alpha$ be defined on some interval $I$ and define for $t_0\in I$ ...
3
votes
1answer
45 views

How to show that $f : \mathbb{R}^n → \mathbb{R}^n$, $f(x) = \frac{h(\Vert x \Vert)}{\Vert x \Vert} x$, is a diffeomorphism onto the open unit ball?

Could anyone help me with the following problem? The problem Fix $\varepsilon \in (0, 1)$ and choose a smooth function $h$ on $[0,\infty)$ such that $h'(t) > 0$ for all $t ≥ 0$, $h(t) = t$ for ...
2
votes
1answer
41 views

immersion of punctured torus in plane

Let $S^1 \times S^1$ be the $2$-torus. If a point $a=(p,q)$ of the torus is removed, i.e., it is punctured at one point then how can I show that it can be immersed in the plane, i.e., in $\Bbb{R}^2$? ...
0
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0answers
28 views

Green's theorem via Stokes's theorem

I am considering the following form of Stokes's theorem: Let $\omega$ be an $n-1$ differential form with compact support on an oriented manifold of dimension $n$. Let us consider the boundary ...
2
votes
1answer
38 views

How to proof that bracket of two vector field can be computed by second derivation

Can some one give a hint how can I proof that where $\phi$ indicated the flow of vector fields.
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29 views

Osculating Circle in Differential Calculus

I am working with an osculating circle as the curve of closest contact to a curve in differential calculus and my book takes some confusing steps that I do not understand. It says: Let $f(x)$ be the ...
3
votes
1answer
30 views

Why not every homogeneous manifold is parallelizable?

It is obvious that not every homogeneous manifold is parallelizable (take for example the two-sphere $S^{2}$). In contrast, every Lie group $G$ is parallelizable, as you can construct a pointwise ...
2
votes
1answer
48 views

Compute Euler characteristic of a compact manifold

We have the manifold embedded in $\mathbb{R}^4$ given by $$M=\{(x,y,z,w)|2x^2+2=2z^2+w^2,3x^2+y^2=z^2+w^2\}$$ How could I compute the Euler characteristic? I've no idea computing the homology group of ...
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vote
1answer
27 views

Cartan's structural equation

I am reading through a proof of Cartan's Structural equation: $$\Omega=d\omega + \frac{1}{2}[\omega\wedge\omega]$$ In the case when the input is two vertical vectors $V_1$ and $V_2$, we can ...
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0answers
34 views

Pullback 1-differential form

Let $(x_0,x'_0) \in \mathbb{R}^2$ be initial data for the Euler Lagrange equation with some given Lagrangian $L: \mathbb{R}^2 \rightarrow \mathbb{R}.$ Then $F^t$ is the flow that maps the initial ...
3
votes
0answers
48 views

Momentum a cotangent vector?

Imagine we have a particle described by $x \in M$, where $M$ is some manifold, then it is very intuitive I think that a velocity is an element of the tangent space at $x$, so $x' \in T_{x}M.$ Thus, by ...
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1answer
40 views

Proving that these curves intersect

Let $\Gamma$, $\Sigma$ be two curves with ranges in $(\{0\}\cup\mathbb{R}_{+})^2$. $\Gamma$ starts on the $y$ and ends on the $x$ axis: $\Gamma(0)=(0,\gamma_2),\Gamma(1)=(\gamma_1,0)$. $\Sigma$ is a ...
2
votes
1answer
42 views

Question about parallel displacement on a surface

This is Problem 9.6(1) from the book The Geometry of Physics: What's wrong with the following argument? A vector $\mathbf v$ is parallel displaced around a small closed curve $C = ...
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0answers
18 views

Twice differentiable functions that are harmonic

This is a question that I have spotted in a textbook for differential geometry. Determine all twice differentiable non-zero functions g : R $\rightarrow$ R and h : R $\rightarrow$ R such that $f ...
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2answers
31 views

Möbius strip as a non-trivial principal bundle

There is a well-known theorem that a principal bundle is trivial if and only if it admits a global section. I'm trying to get a good picture of what this theorem means. The Möbius Strip can be ...
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0answers
28 views

On critical values of a linear function $g:SO_2\times \mathbb R^2\to \mathbb R^2$

Consider map $g:SO_2\times \mathbb R^2\to \mathbb R^2$, $g(A,v)=Av$, where $A\in SO_2$ is an orthogonal $2\times 2$ matrix and $v\in \mathbb R^2$ is a $2$-vector. Show that $0$ is a critical value. ...
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0answers
34 views

Self-adjointness of $X^2$

Let $X$ be a vector field on a compact (may be even complete) Riemannian manifold without boundary. I am wondering if $X^2$ will be a self-adjoint operator on $L^2(M)$. Any hints would be appreciated. ...
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0answers
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Unbounded Geodesics and Nonpositive Curvature

I have the following interesting(?) question: Let $M$ be a complete Riemann manifold. Suppose that it is of non-negative curvature. I want to know if $M$ has unbounded geodesics. As the question is ...
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0answers
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Fiber summation of symplectic Lefschetz fibrations.

Recall the following standard result. Theorem: For any genus $2$ symplectic Lefschetz fibration $f:X\to S^2$, there exists an integer $n_0$ such that, for all $n\ge n_0$, $f\# nf_0$ is isomorphic ...
3
votes
0answers
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Total curvature of an ovaloid.

I have the following exercise that I have to solve without using Gauss-Bonnet theorem. We say that a compact surface $\Sigma \subset \mathbb{R}^3$ is an ovaloid if the Gaussian curvature $K(p)>0$ ...
2
votes
0answers
25 views

Write down the explicit form of the $15$ Killing vectors of the 5-sphere

I am looking for a way to write down explicitly the $15$ vectors which are generators of $SO(6)$ in polar coordinates on the $5$-sphere. In particular I have the round metric $$g_{\mu\nu} = \left( ...
0
votes
1answer
23 views

Jets and vertical differential

For a vector bundle $(E,\pi, M)$ let $\phi :M\mapsto E$ be a section of $\pi $, $x\in M$ and $u=\phi (x)$. The vertical differential of the section $\phi$ at point $u\in E$ is the map: ...
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0answers
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Non-affinely parametrized geodesics

Consider a non-affinely parameterised geodesic, i.e., a geodesic whose tangent vector field obeys $\nabla_X X = fX$ for some function $f$. Prove that one may reparameterise the geodesic so the tangent ...
2
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0answers
37 views

How to prove that the tangent map $T\phi$ into the pullback bundle is smooth?

Assume $\phi: M\rightarrow N$ is smooth. Let $\phi^*(TN)$ be the pullback bundle of $TN$ by $\phi$. Define $T\phi:TM\rightarrow \phi^*(TN)$ as follows: $T\phi(m,v)=(m,d\phi_m(v)) $. We also have the ...
3
votes
0answers
33 views

What does a skew second fundamental form geometrically mean?

Could there be a realizable 2-surface in some higher dimensional non-Riemannian embedding space whose second fundamental form is skew? If yes, then what would its skew part mean?
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0answers
14 views

Asymptotic geodesic on hyperboloid.

Consider a geodesic which starts at a point $p$ in the upper part $(z>0)$ of a hyperboloid of revolution $x^2+y^2−z^2=1$ and makes an angle $\theta$ with the parallel passing through $p$ in such a ...
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votes
1answer
171 views

Different definitions of differential forms?

I am a physicist and was reading about differential forms in Classical Mechanics. Now, I thought that a two-form is a smooth map $\omega : M \rightarrow \Lambda(T^*M)$ so that a point $p$ on the ...
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Bianchi Identity - Gauge Theory

I am reading through some lecture notes (found here) and following a proof of the Bianchi identity in the context of principal bundles. That is, $h^*\Omega = 0$, where $\Omega$ is the curvature ...
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votes
1answer
27 views

Curve Orientation on a Surface

Let $S\subset\mathbb{R}^3$ be an orientable surface, but not oriented (yet). Let $X:(u_1,u_2)\in U\subset \mathbb{R}^2\longrightarrow X(U)\subset S$ be a local parametrization of the surface $S$. We ...
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0answers
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product metric on Riemannian manifolds

Let $M_1$ and $M_2$ be Riemannian manifolds and consider the cartesian product $M_1 \times M_2$ with the profuct structure .Let $\pi_1: M_1 \times M_2 \to M_1$ and $\pi_2: M_1 \times M_2 \to M_2$be ...
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votes
1answer
97 views

Differential forms - looking for 3 definitions!

I am sorry for this type of question, but I currently have to deal with differential forms although I have not heard so far what they actually are, so I have just a few very particular questions about ...
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vote
1answer
23 views

Parabolic points and curvature.

I have problems to solve this exercise: Let $p$ be a point of an oriented surface $S$ and assume that there is a neighborhood $U$ of $p$ in $S$ all points of which are parabolic. Prove that the ...
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votes
0answers
18 views

Frenet frame and tangent space.

Let $\gamma(s) \subset \Sigma \subset \mathbb{R}^3$ a parametrized curve by arc lenght. Let suppose that $\gamma$ is on an oriented surface $\Sigma \subset \mathbb{R}^3$. We can consider the Frenet ...
2
votes
1answer
48 views

Diffeomorphism between a sphere and ellipsoid in $\mathbb R^3$.

Diffeomorphism between a sphere and ellipsoid in $\mathbb R^3$. I've been looking for an diffeomorphism between a sphere in $\mathbb R^3$ and an ellipsoid of the form $$\{ (x,y,z) \in \mathbb R^3 ...
0
votes
0answers
31 views

Deformation theory in a paper of Bogomolov and Tschinkel

I am trying to read this paper http://www.math.nyu.edu/~tschinke/papers/yuri/00ajm/ajm.pdf, by Bogomolov and Tschinkel. I had 0 preparation in the theory of deformation of complex structures, but in ...
3
votes
1answer
64 views
+100

Hausdorff measure, volume form, reference

Could you tell me where I can find a reference to the fourth corollary in this encyclopedia? Corollary $4$: Assume that $\Sigma \subset \mathbb{R}^m$ is an $n$-dimensional $C^1$ ...