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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integrating by parts on a manifold

Suppose $M$ is compact. Let $\phi$ be some smooth function, and $\beta$ an $n-1$-form. Then does integration by parts say that $$\int_M\phi d\beta=\int_Md\phi\wedge\beta?$$ If not, how does ...
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70 views

Motivation for the definition of an infinitesimal object

An infinitesimal object $D$ in a Cartesian closed category $\mathsf{C}$ is one for which the internal Hom functor $$(-)^D: \mathsf{C} \to \mathsf{C}$$ has a right adjoint. I am wondering what is the ...
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1answer
36 views

Kähler–Einstein condition

Let $(M,\omega)$ be a Kähler manifold, and $g$ the Riemannian metric such that $\omega(X,JY)=g(X,Y)$. If there is a function $f$ such that $\operatorname{Ric}=f\omega$ does this mean that ...
2
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1answer
28 views

Dilating a curved ball

Let $B$ be a ball sitting inside a manifold $(M^n, g)$. Now, let us dilate the metric $g$ to $\lambda g$, $\lambda$ being a positive number going to $\infty$. It seems intuitively true that the ...
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24 views

If $\Gamma\subseteq Diff(M)$ is finite dimensional, when is the evaluation $\Gamma\rightarrow M$ a submersion?

Let $M$ be a smooth manifold. Say it is compact and connected. Suppose that there exists a finite dimensional submanifold $\Gamma\subseteq\mbox{Diff}(M)$ such that the evaluation map ...
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75 views

Is a variety a CW-complex?

How to establish that any differentiable manifold and any complex algebraic variety is a CW-complex ? Thank you in advance for your help.
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52 views

Connection on Tangent Bundle of Group Manifolds

The thing puzzling me is about a transformation rule of connection on tangent bundle of group manifolds: Assume one has a compact and simply connected Lie group $K$. One can give a metric $g$ on $K$ ...
2
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1answer
20 views

differentiable structure on mobius strip

Define $M= \mathbb{R}^2/\sim$ where $(x,y)\sim(x',y')$ if $x-x'=2n$ for some integer $n$ and $y = (-1)^n y'$. Then how can I give a differentiable sturucture on $M$? Is there a general technique for ...
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30 views

Extending a function on a submanifold to the ambient manifold & proof of a property of a vector field.

$\newcommand{\wt}[1]{\widetilde{#1}}$ Hello, I just tried my hand at two exercises from John M Lee's book Riemannian Geometry and I would like to know whether my reasoning is sound or if I did ...
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1answer
52 views

Another Differential Geometry-Curve Theory

This is another problem that keeps arising year after year that none is able to solve. Any help is very appreciated. Let $r(s)$ be a regular closed curve which lays in sphere $S^2$. Prove that: ...
2
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1answer
60 views

Differential-Geometry question- Curve Theory

Let $r(s)$ be a curve parametrized by the natural parameter $s$ and for its curvature $k$ and torsion $t$ the following condition applies: $$k(s),t(s)\neq 0 $$ for every $s$. Prove that the curve ...
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Integration over manifolds

S is subset of $\Bbb R^3 $ consisting of the union of 1) $z$ axis 2) the unit circle $x^2+y^2=1,z=0$ 3) the points $(0,y,0)$ with $y \ge1 $ Let $A$ be the open set $\Bbb R^3-S$. Let $C_1, C_2, ...
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56 views

Is there a fibre bundle with fibre homeomorphic to $\mathbb R^k$ which cannot be given the structure of a vector bundle?

We define a (rank $k$) vector bundle to be a fibre bundle with fibre $\mathbb R^k$ such that the local trivializations are fibre-by-fibre vector space isomorphisms. I'm curious whether this linearity ...
5
votes
1answer
76 views

What is the geometric interpretation of the Koszul formula?

I saw this simple form of Koszul formula on a book: $$2\ g(\nabla_XY,Z) = \mathcal{L}_Yg(X,Z) + (d\theta_Y)(X,Z)$$ where $\theta_Y$ is the one-form $g(Y,\cdot)$. It is equivalent to the more ...
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19 views

Metric, torsion free connections on principal bundles

Let $F(M)$ be the frame bundle of a $n$-dimensional differentiable manifold $M$, and let $H\subset TF(M)$ be a connection. A Riemannian metric on $M$ can be equivalently written as an equivariant ...
2
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1answer
40 views

a tangent vector which belongs to intersection of a manifold and a subspace is tangent to their intersection?

I have the following situation: There is an embedded submanifold (in my particular case it is of codimension one but I do not think it matters here) $M\subseteq R^n$, and a vector subspace ...
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2answers
57 views

$SL(3,\mathbb{R})$ is a smooth manifold?

How do you show $SL(3,\mathbb{R})$ is a smooth manifold? I am thinking to use the preimage theorem, but what kind of thing I need to show first before I can apply the theorem?
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22 views

Trouble proving identity - Gauge theory/Maurer-Carton one-form/Adjoint representation

The Identity I am trying to prove is the one in this already asked question how to show that ${ad}_{g_{\alpha\beta}} \circ g_{\alpha\beta}^{\star}\theta=-g_{\beta\alpha}^{\star}\theta$? The author ...
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2answers
49 views

Geometric characterization of critical points of the Gauss map.

Let $\Sigma \subset \mathbb{R}^3$ an oriented surface by Gauss map $N: \Sigma \rightarrow S^2$. How can I find a geometric characterization of critical points of $N$?
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1answer
165 views

Intuitive meaning of immersions

I have a hard time understanding the concept of immersions. In my course, it was only introduced by the immersion theorem wich says: Let $f: U \subset \mathbb{R}^{m} \rightarrow \mathbb{R}^{n}$ be ...
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1answer
55 views

Finding critical values of a function on an embedded surface

Prior to the problem, we have already shown that $\Sigma=\{x_1x_2^2+x_2x_3^2+x_3x_1^2=1\}\subset\mathbb{R}^3$ is an embedded hypersurface and that the function ...
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1answer
34 views

Is there a general coordinate transformation perserving the components of an Euclidean metric?

In the Euclidean space (or Lorentz spacetime, if you are interested in relativity), there is one orthonormal coordinate system $\{x^\mu\}$ such that the distance squared is given by ...
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Characterize the sphere using mean curvature.

We know the following result: if $\Sigma$ is a compact surface than $$ \int_{\Sigma}H^2 \ge 4 \pi, $$ where with $H= \frac{1}{2}(\kappa_1+\kappa_2)$ we denote the main curvature. I have to prove that ...
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Integration over a variety

If $ M $ is a differentiable manifold equipped with an Atlas $ \mathcal{A} = ( U_i , \varphi_i )_{ i \in I} $, we can then calculate the integral of a differential form $ \omega $ over $ M $ with the ...
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39 views

Generalize Gauss-Bonnet Formula to non-simple closed curves

According to the Classical Gauss-Bonnet Formula, I think it should can be generalized to non-simple closed curves in the following sense: For a domain $\Omega$ enclosed by an non-simple closed curve ...
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2answers
71 views

Derivative of determinant at some point

Let $c:\mathbb{R} \rightarrow \mathbb{M}_n(\mathbb{R})$ defined by $$c(t)=A e^{tB}$$ where $A\in GL(n,\mathbb{R})$ and $B \in \mathbb{M}_n(\mathbb{R})$. The question ask me to find $c'(0)$ and ...
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36 views

Geometric interpretation of Gaussian curvature.

We have the following result from Do Carmo book of differential Geometry: "Let $p$ be a point of a surface $\Sigma$ such that the Gaussian curvature $K(p) \neq 0$, and let $V$ be a connected ...
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3answers
34 views

Primitive of an Exact form

I have the following differential $3$-form, $$ H = \cos \zeta \sin \zeta\, \mathrm{d}\zeta \wedge \mathrm{d}\varphi_1 \wedge \mathrm{d}\varphi_2 $$ which I know is exact by the properties of my ...
3
votes
1answer
41 views

Want to show two smooth manifolds are diffeomorphic

Consider a smooth manifold $M = \{ (u,v) \in \mathbb{R^3} \times \mathbb{R^3} \mid \|u\|=\|v\|=1 \text{ with } u \perp v \}$, and want to show $M$ is diffeomorphic to $SO(3)$, the rotational group in ...
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1answer
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Extension of Sections of Restricted Vector Bundles

Edit: Changing Question: There are two questions related questions: extending a smooth vector field extending a vector field defined on a closed submanifold I'm trying to answer a question which is ...
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About the associated bundle of the bundle of affine frames $A(M)$

According to Kobayashi & Nomizu (Foundations of differential geometry vol.1) the associate bundle for bundle of linear frames $L(M)$ is the tangent bundle $TM$ since $GL_n(\mathbb{R})$ acts freely ...
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40 views

A question about stereographic projection of a plane onto a sphere

I am reading a paper by Christine Bernardi, available here http://epubs.siam.org/doi/pdf/10.1137/0726068, my question relates to page 1237, which I shall elaborate on: In this part we have the unit ...
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1answer
55 views

Computing the differential

Assumption(s): Let $(\rho, M(n,\mathbb{C}))$ be a representation of the group $G = SU(n)$ where $\rho(g)A := gAg^{T}$ for $A\in M(n,\mathbb{C}),g\in G$ Problem: I want to compute the ...
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Space of invariant sections

I have a principal bundle $\pi: M \to B$ with structure group $G$ and a vector bundle $p: V\to M$. I need to work with $\Gamma_G(V,M)$, the space of invariant (under the fiber action on $M$) sections ...
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1answer
42 views

Lower bound on convexity radius in terms of injectivity radius (without using curvature)

Let $M$ be a complete Riemannian manifold, and let $C$ be a subset of $M$. We will say $C$ is convex if for any points $p,q \in C$, there exists a unique normal minimal geodesic $\gamma$ joining $p$ ...
2
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1answer
46 views

Does the Riemann tensor encode all information about the second derivatives of the metric?

In answer to this question I suggested the following as a motivation for the definition of the Riemann tensor: Let two $\mathcal M$ and $\mathcal N$ be two dim-$n$ Lorentzian manifolds with ...
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26 views

Pappus theorem and area of a revolution surface.

Let $y=f(z)$ be a function. How can I calculate the area of the surface obtained rotating the function along the $z$ axis, where y is the revolution torus? Is it possible to do it using Pappus formula ...
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28 views

Proving path of motion is a Geodesic in general reletivity.

I am studying the work of Miguel Alcubierre, in particular his warp drive metric. A consequence of his metric is that the ship will travel on a geodesic and this is what I am trying to prove. I ...
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2answers
56 views

How do I show $SO(n)$ is open and closed in $O(n)$?

As title, how do I show $SO(n)$ is open and closed subset of $O(n)$? Is the preimage of closed set under continuous map closed ?
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21 views

Conformal group, Minkowski metric

The conformal group is the subgroup of coordinate transformations that leave the metric invariant up to a scale factor. So under a transformation $x\rightarrow x'$ we have $g_{\mu\nu}(x)\rightarrow ...
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18 views

Why is the kernel of the connection one form a connection on a principal bundle?

Let $\pi:P\rightarrow M$ be a principal bundle and let $\omega\in \Omega(P;\mathfrak{g})$ be a one form satisfying $\omega(\sigma(X))=X$ and $R_g^*\omega=\text{Ad}_{g^{-1}}\circ\omega$ Then ...
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26 views

Construction of a diffeomorphism handling varying domain

Let $\Omega$ be a strictly convex domain, $\partial\Omega\in C^{2,1}$ We define a foliation $\{\Omega_t\}_{0\leq t\leq 1}\subset\Omega$ as follows. Let $\Omega_0=B_r(x_0)$, a small ball centered at ...
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Manifolds with smooth structure

One of the remark in my lecture notes said: In dimension $\leq 3$, every topological manifold has a unique smooth structure (up to diffeomorphism.) I don't quite understand what is a structure ...
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2answers
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Showing a limit is equivalent to the curvature of a planar curve

I've been working on this limit for a long time and just don't know how to show this. I tried representing h and d as vectors, but I cannot seem to accurately describe them. What would be a good ...
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1answer
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An exercise in Do Carmo's book-Differential geometry of curves and surfaces

In do Carmo's book (page 11, exercise 9), the author gives an example of $C^0$ unrectifiable curve: Let $\alpha: [0,1]\to \mathbb R^2$ be given as \begin{equation*} \left\{\begin{aligned} \left(t, ...
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Normal Curvature $κ_n(p, v)$

Let $S$ be the embedded torus with parametrization $σ(θ, ϕ) = ((2 + \cos θ) \cos ϕ,(2 + \cos θ) \sin ϕ,\sin θ)$. The first and the second fundamental forms of $σ$ are $dθ^2 + (2 + \cos θ)^ 2 dϕ^2$ ...
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71 views

Flow of a gradient vector field

I would like to know how the flow of a gradient vector field of a function $f$ (without critical points), on a riemannian manifold, behaves in relation with the level sets. I mean when does it take ...
2
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0answers
35 views

Difference of two connections is a tensor

I am currently reading through Jost's Riemannian Geometry and Geometric Analysis and am seeking clarification of the statement in the title. Jost abstractly defines a connection on a vector bundle ...
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1answer
29 views

Does the dual basis to some basis of $T^*_pM$ looks localy like a coordinate chart?

Let $M$ be a manifold and let $\{\alpha_k\}$ be a set of $1$- forms s.t. $\{\alpha_k(p)\}$ forms a basis for $T^*_pM$. Let $(x,U)$ be a chart based in $p$ and denote $\partial_i := ...
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41 views

Differential Geometry - Distributions mutually orthogonal, span the tangent space, parallel imply manifold splits locally as product manifold

I'm stuck on a portion of Exercise 21, Chapter 2 in Petersen's Riemannian geometry text. Fix a Riemannian manifold $(M,g).$ Suppose that I have two distributions $D^1$ and $D^2$ defined on $M.$ ...