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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266 views

Orientability of the total space of a vector bundle over an oriented manifold

Let $M$ be a (smooth) manifold of dimension $n$, and let $\pi : E \to M$ be a (smooth) vector bundle of rank $r$. If I choose a connection on $E$, then I obtain a decomposition of $T E$ as $V E ...
6
votes
1answer
272 views

Are these definitions of submanifold equivalent?

Let $M$ be a manifold – by which I mean a second countable Hausdorff smooth manifold. Here's an "obviously" correct definition of (embedded) submanifold: Definition A. A subspace $S$ of $M$ is a ...
4
votes
1answer
228 views

Umbilic points on a connected smooth surface problem

Let $S\subset \mathbb{R}^3$ be a connected smooth surface. Suppose that every point of $S$ is an umbilic point. Prove that $S$ is a subset of either a plane or a sphere in $\mathbb{R}^3$. Here's a HW ...
8
votes
1answer
185 views

How to prove $(0,1)$ form is not $\overline\partial$-exact

On a complex manifold, if we are dealing with the $d$ operator, there's a pretty easy way of showing some form is not $d$-exact, simply by integrating in a closed loop. If you can find a loop that is ...
2
votes
3answers
220 views

Proposition about curves in $S^2$

Let $\gamma_1,\gamma_2:(a,b)\to S^2$ be unit speed curves in $S^2=\{\vec{v}\in\mathbb{R^3}:\vec{v}\cdot\vec{v}=1\}$. Then the following two statements are equivalent: (1) There is a $3\times 3$ ...
2
votes
1answer
411 views

Properly discontinuous action

This is exercise 9 of chapter 0 from Do Carmo's book in Riemannian Geometry: Let $G\times M \rightarrow M$ a properly discontinuous action from a Group $G$ on a smooth manifold $M$. Prove that ...
0
votes
1answer
389 views

The conditions of a metric to be geodesically complete

On $\{\vec{x}\in \mathbb{R^n}:x_1^2+x_2^2+\cdots+x_n^2<1\}$ in $\mathbb {R^n},$ what $\alpha$ can make the metric $$g=(1-x_1^2-x_2^2-\cdots-x_n^2)^{-\alpha}(dx_1\otimes dx_1+dx_2\otimes ...
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0answers
154 views

Isometry from a surface to itself.

We are talking about surfaces in $\mathbb{R}^3$ here. I know that not every isometry from a surface to another is a congruence. But what about isometries from a surface to itself? Can someone give ...
2
votes
3answers
102 views

How do I apply smooth $p$-forms to vector fields?

Let $M$ be a smooth manifold. My definition of a smooth $p$-form is a map section $\omega: M \rightarrow \Lambda^p TM^*$, i.e. if $q \in M$ is contained in a chart $U$ with co-ordinates $x_1, \ldots, ...
0
votes
1answer
802 views

Surface gradient on unit sphere

Let $\Gamma$ be the unit 2-sphere, say and let $f:\Gamma \to \mathbb{R}$ be some nice function. My teacher says when i calculate the surface gradient $$\nabla_\Gamma f = \nabla f - (\nabla f\cdot ...
0
votes
1answer
40 views

Path contained in Surfaces

$y(t)$ is a path contained in two surfaces: $x^2+y^4+z^6=3$, $x+y^2=y+z^2$ also $y(0)=(1,1,1)$ and $||y'(0)||=1$ Need to find the vectors $-y'(0)$ and $+y'(0)$ To be honest, I'm not sure how to ...
7
votes
1answer
292 views

Most succinct proof of the uniqueness and existence of the Levi-Civita connection.

Seeing as proving the existence and/or uniqueness of the Levi-Civita connection seems to crop up in every single exam in Geometry and General Relativity, what is the most succinct proof of this, to ...
0
votes
2answers
126 views

The rank of mapping on tangent bundle

$f$ is a mooth mapping from differential manifold $M$ to $N$ and $T(f)$ is the induced mapping on their tangent bundles. For $x\in M$, does ${\rm{ran}}{{\rm{k}}_{{h_x}}}(T(f)) = ...
7
votes
1answer
338 views

Metric on $\Bbb{R}^n$ which comes from a continuous function

I've been struggling to prove the following fact for some time now, and I didn't manage to do so. Suppose $W : \Bbb{R}^n \to \Bbb{R}_+$ is a continuous, positive function, with exactly $n$ zeros ...
1
vote
2answers
2k views

What is the shortest path equation between 2 points on a cone?

What is the shortest path equation between 2 points (from A to B) on a cone surface? $A= (x_1,y_1,z_1)$ and $B=(x_2,y_2,z_2)$ and cone equation is $x^2+y^2=r^2z^2$ I know that the shortest path ...
1
vote
2answers
409 views

Riemann tensor on a sphere

I was given this exercise but I don't even know where to start: to compute the Riemann tensor of the 2-dimensional sphere. The tensor acts on vector fields X,Y,Z like this: ...
1
vote
1answer
85 views

Signature and Orientation

It is clear that a reordering of the elements in a chosen basis for an n-dimensional vector space induces a permutation on n elements, and conversely such a permutation corresponds to a re-ordering of ...
4
votes
2answers
1k views

Finding parametric curves on a sphere

Is there some general method for finding such curves? Let's say I have a planar curve, how can I project it onto a sphere? I am interested in a curve that starts at the south pole of a sphere, then ...
1
vote
0answers
61 views

How to find the boundary of a $\mathcal{C}^1$ manifold?

Let $U$ be a bounded open convex set of $\mathbb{R}^d$, and $\Phi$ a differentiable map from $U\times \mathbb{R}$ to $\mathbb{R}^d$. The object of interest for me is $R=\Phi(U\times \mathbb{R})$. The ...
16
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1answer
2k views

is there any good resource for video lectures of differential geometry?

I am wondering if there is some online resource for video lectures on the topic of differential geometry. Thanks a lot
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1answer
591 views

Help in proving: even dimensional Real Projective space is not orientable

Prove that an even dimensional Real Projective space is not orientable.
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1answer
86 views

The extension of function

$M$ is a smooth manifold and $A$ is a subset of $M$. If a function $f$ on $M$ is $C^\infty$ on $A$(that is, for every point $x\in A$, there is a open set $V_x$ and a $C^\infty$ function $f_x$ such ...
3
votes
1answer
687 views

orientation preserving map

Let $f:X\rightarrow Y$ be a diffeomorphism between connected oriented manifolds. $f$ is orientation-preserving at $p\in X$ if the induced map $df_{p}:T_{p}X\rightarrow T_{f(p)}Y$ is ...
0
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1answer
188 views

Locally Euclidean

A quick definition check: Would someone please tell me what "locally Euclidean" means when applied to a Riemannian metric? I have kind of an idea about it, for example when we multiply by a ...
6
votes
1answer
176 views

Is there a fundamental misunderstanding here or have I made an algebraic slip?

Is there a fundamental misunderstanding here or have I made an algebraic slip? I have a Riemannian metric of the form $ds^2={du^2+dv^2\over 1-u^2-v^2}$ on an open disc and I want to prove that radial ...
4
votes
0answers
210 views

How to prove the holonomy group is preserved under Ricci flow?

I've heard that on a Kaehler manifold $(M,g_0)$, if you evolve the metric $g$ by Ricci flow $\partial g_{ij}(t)/\partial t=-2R_{ij}$, and $g(0)=g_0$, then you always have $g(t)$ is a Kaehler metric on ...
3
votes
1answer
533 views

Laplace-Beltrami operator for curves

I'm CS major and have used discrete Laplace-Beltrami operator for 2D-manifold (surface meshes). I'm wondering if it is possible to define Laplace-Beltrami operator for 1D-manifold. If this is ...
3
votes
0answers
82 views

Closed / embedded surface

Given a closed surface in $\mathbb R^3$, is it necessarily an "embedded surface"? I think it is true, but that is just because I can't think of a closed surface for which we cannot construct a smooth ...
2
votes
1answer
378 views

Is the union of two manifolds a manifold?

Suppose I have $M$ and $N$, two $k$-manifolds in $\mathbb{R}^n$. Is it true that $M\cup N$ is also a manifold? What is a sufficient condition for positive answer?
1
vote
1answer
67 views

to show $g$ attains maxima and minima

Let $A$ be a symmetric $n\times n$ real matrix and define $G:\mathbb{R}^n\rightarrow \mathbb{R}$ by $G(t)=\langle At,t\rangle$; let $g:S^{n-1}\rightarrow \mathbb{R}$ be the restriction of $G$ to the ...
2
votes
1answer
124 views

Construction of manifold

let $\Omega$ be an open subset of $\mathbb{R}^n$ and $f$ be a smooth real valued function on $\Omega$. Let $M=\{p\in\Omega:f(p)\ge 0 \}$, and $\Gamma=\{p\in\Omega :f(p)=0\}$. Suppose $M\neq \phi$ and ...
2
votes
1answer
150 views

ellipticity of the Laplacian associated to the de Rham complex

I am struggeling with the following comment that I read regarding the de Rham complex: Define $(d + \delta)_e : C^\infty(\Lambda^e(T^*M)) + C^\infty(\Lambda^o(T^*M))$ where \begin{equation} ...
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0answers
89 views

Different definitions of the complex Grassmannian

I have come across two different definitions of what I suspect is the same object. Both are called the complex Grassmannian: 1: $U(n)/U(k)\times U(n-k)$ 2: $SU(n)/(S(U(k)\times U(n-k))$ What is the ...
3
votes
1answer
167 views

The diffeomorphism of $\mathbb R^n$

If $f$ is a diffeomorphism of $\mathbb R^n$ and $K$ is a compact set in $\mathbb R^n$, can we find another diffeomorphism $\tilde f$ of $\mathbb R^n$ such that: (1)$f=\tilde f$ on a neighborhood of ...
0
votes
1answer
294 views

diffeomorphism of derivative map at tangent space level

$f: X\rightarrow Y$ is a diffeomorphism, then at each $x$ its derivative $df_x$ is an isomorphism of tangent spaces.could you please give me proof and insight of this result?
2
votes
0answers
598 views

Pulling back vector fields

I want to find conditions under which one can pull-back vector fields (if it is at all possible). Let $F:M \to N$ be a smooth surjective map between two $C^{\infty}$ manifolds of the same dimension. ...
9
votes
2answers
286 views

Showing that some symplectomorphism isn't Hamiltonian

I have the next symplectomorphism $(x,\xi)\mapsto (x,\xi+1)$ of $T^* S^1$, and I am asked if it's Hamiltonian symplectomorphism, i believe that it's not, though I am not sure how to show it. I know ...
3
votes
2answers
377 views

divergence of a vector field on a manifold

I've been asked to show the following: For a vector field $V$ on a semi-Riemannian manifold with metric $g$ that $$Div \cdot V = \frac{1}{\sqrt{\det(g)}}\partial_i\left(\sqrt{\det(g)}V^i\right)$$ I ...
3
votes
3answers
270 views

Scalar product on manifold.

Let $M$ be a closed Riemannian manifold and $\omega$ and $\eta$ two differential forms of the same degree. Then one can consider $\int_M \omega \wedge *\eta$, where $*$ denotes the Hodge star ...
4
votes
2answers
751 views

About connected Lie Groups

How can I prove that a connected Lie Group is generated by any neighborhood of the identity? The result is almost trivial for $R^n$ but I tried using the open subgroup generated by this ...
2
votes
1answer
99 views

Did I integrate a differential form correctly?

I start with 1-form $\omega=f\,dx$ on $\left[0,1\right]$ where $f\left(0\right)=f\left(1\right)$ and a $g:\left[0,1\right]\to R$ with $g\left(0\right)=g\left(1\right)$ and I want to integrate ...
4
votes
1answer
346 views

Stokes theorem for Lorentz manifolds

Reading Tao's book: Nonlinear Dispersive Equations I came upon an identity (the energy flux identity for the wave equation, page 90) for which the proof uses the Stokes theorem. In this case he uses ...
2
votes
1answer
115 views

Translating theorem on closed/exact forms

I'm trying to translate this theorem, below, into theorems about scalar and vector fields in $\mathbb R^3$: Theorem: Let $A$ be a star-convex open set in $\mathbb R^n$. Let $\omega$ be a closed ...
0
votes
0answers
119 views

Derivative/Chain Rule (for MANLYfolds) Computation

Embarrasingly, I can't compute the following derivative. $dh(X)=\left.\frac{d}{dt}h(e^{XT}]\right|_{t=0}$, where $X$ resides in the lie algebra of $\rm SL(3,\Bbb C)$ [ie $\mathfrak{sl}(3,\Bbb C)$] ...
2
votes
1answer
141 views

DeRham Cohomology

Let $p$ and $q$ be two points of $\mathbb{R}^n$ where let $n\geq 1$. Then $$\dim H^k(\mathbb R^n - p - q) = \begin{cases}0, &\text{ if }k\text{ is not equal to }n-1,\\ 2,&\text{ if }k = ...
3
votes
2answers
413 views

Lie Algebra Homomorphism Question

So this is a bit of a follow-up to my recent question. I don't mean to inundate the feed with my quandaries, but as I move through the theory I keep hitting stumbling blocks (which y'all so kindly ...
1
vote
1answer
169 views
2
votes
2answers
148 views

Does a non-zero wedge product make a coordinate system?

I already know that a coordinate system must have a non-zero wedge product of the components, but does it go the other way, that is, does $df_1\wedge\cdots\wedge df_n(p)\ne0$ mean that $f$ is a ...
0
votes
1answer
78 views

Transforming vectors to coordinate vectors

Let $X_1,\dots,X_n$ be n vector fields on an open subset $U$ of a manifold of dimension $n$ Suppose that at $p\in U$, the vectors $(X_1)_p,\dots,(X_n)_p$ are linearly indipendent. would any one say ...
1
vote
1answer
115 views

group of automorphism of a vector space

While reading manifold theory I stuck to this problem: $V$ be a vector space with $\dim V<\infty$ over $\mathbb{R}$ and $GL(V)$ be the group of all linear isomorphisms of $V$ into itself. A basis ...