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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Tangent and normal vectors for parameterization of a straight line.

Hi everyone just a simple question. Suppose the curve $$r(t) = (x_0, t)\\ 0 \leq t \leq 1 $$ $x_0$ is some real number. We see that the unit tangent $T(t)= (0,1) = \dfrac{r'(t)}{\|r'(t)\|} $ ...
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1answer
50 views

Vanish christoffel symbols implies $g_{ij,k}=0$

Let $(M,g)$ be a pseudo-riemann manifold and $(U,\psi=(x^1,\ldots,x^n))$ a local chart around some point $p$ in $M$. It is easy to show that if $\partial g_{ij}/\partial x^k=0$ in $p$ for all $i,j,k$ ...
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34 views

Lorentz manifolds - Change of co-ordinates

Let $(M,g)$ be a lorentz 4-dimensional manifold. Let $p\in M$ and $(U,\psi=(x^1,x^2,x^3,x^4))$ a local chart around $p$ such that $x^i(0)=0$ for all $i$. Let new coordinates $\bar x^i$ that verify: $$...
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0answers
35 views

Sectional curvature of 2-manifold

This is problem7, chapter 5 of Do Carmo's Riemannian geometry. Let $M$ be a Riemannian two manifold, $p\in M$ , $exp_p$ is a diffeomorphism on a neighbourhood of origin $V\in T_pM$. Let $S_r(0)\...
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20 views

How is the Euclidean mean curvature of a minimal submanifold of $ \mathbb{S^{n-1}} $ is equal to the metric Laplacian of the position vector?

I am reading about minimal cones from the book "A Course in Minimal Surfaces, T.H Colding, W.P Minicozzi II". It says that if $N^{k-1} \subset \mathbb{S^{n-1}}$ is $k-1$ dimensional minimal ...
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1answer
31 views

Half strip neighbourhoods for regular surfaces

Let $S$ be a regular connected and compact surface in $\mathbb{R}^3$. It is well known that such surfaces admit a global tubular neighborhood, of thickness $\epsilon>0$. In particular, by ...
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28 views

Limit of the commutator of two elements?

Given a Lie group $G$ such the $\mathfrak{g}$ denoted its Lie algebra. Let $[g,g']_{G}$ the commutator of two elements $g,g' \in G$ and denoted by $[X,X']_{\mathfrak{g}}$ the Lie bracket of two ...
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42 views

What will happen if evolve metric under Ricci flow on general manifold?

Because the scalar curvature under Ricci flow evolve by $$ \partial_t R=\Delta R+ 2|Ric|^2 $$ I treat it as heat equation with heat source . So, no matter the heat source is very hot or not , the ...
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1answer
47 views

compute tangent map

Let $f:O(n)\rightarrow O(n), f(M)=M^3$ be a map, $O(n)$ are the orthogonal matrices. Calculate the tangent map at $I$. My idea would be to firstly calculate the tangent space at $I$, it is the kernel ...
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31 views

Convergence of Discretized Geodesics?

Let $M$ be a Riemmanian manifold, $p\in M$ and $V\in T_p(M)$. Suppose $f^{-1}:U_p \mapsto \mathbb{R}^D$ is a diffeomorphism of a neighborhood of p to an open subset of $\mathbb{R}$ and define the ...
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Why do we need Hausdorff-ness in definition of topological manifold?

Suppose $M^n$ is a topological manifold, then $M^n$ locally looks like $\mathbb{R}^n$. $M^n$ is locally Hausdorff, since $\mathbb{R}^n$ is Hausdorff and Hausdorff-ness is a topological invariant. ...
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1answer
18 views

Connection after a metric rescaling and compatibility

It's known (see here for example) that after a rescaling of the metric $\tilde{g}=e^{2\omega}g$, we can find a new connection $\tilde\nabla$ associated to the new metric: $ \tilde\nabla _X Y = \nabla ...
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1answer
21 views

Holonomy of a curve in case of principal $U(1)$ bundle

Suppose $\pi : P\rightarrow M$ is principal $U(1)$ bundle. Let $\gamma$ be a loop in $M$ based at $x_0$ and write $iA$ as connection 1-form on $P$ where $A\in \Omega(P)$. Now define $hol_{\gamma}(A)\...
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1answer
21 views

normal connection on immersed hypersurface vanishing

I am studying Riemannian geometry using Do Carmo's book. I am learning about isometric immersions right now, and I got stuck with the following claim about Codazzi's equation. Let $f:M^n \to \...
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2answers
53 views

Ruled surface of constant K gauss curvature

Ruled surfaces have negative /zero K ; so what are some examples, with parametrization, of a ruled surface with constant negative K ? EDIT1: For standard ruled surface we need to integrate linked ...
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2answers
178 views

Lie bracket of coordinate frame

Suppose I have a local nbd $U$ in a manifold centered around $p$ and a chart on it. The chart is given by $X=(x_1,x_2,\cdots,x_n):U\rightarrow V\subset\mathbb{R}^n$. I consider the vector fields $\...
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0answers
90 views

Stereographic projection with de Sitter space and hyperbolic plane

How can we do stereographic projection using de Sitter space $\Bbb S^2_1$ and the hyperbolic plane $\Bbb H^2$, in Lorentz-Minkowski space $\Bbb L^3$. For $\Bbb S^2_1$ it is not clear what point ...
2
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1answer
54 views

Hodge theory on semi-Riemannian manifolds [reference request]

I need to learn a bit of Hodge theory on manifolds and I am looking for a reference which covers the case where the metric has arbitrary signature $(p,q)$. Most books I have found seem to focus on the ...
17
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3answers
654 views

How to prove Lagrange multiplier theorem in a rigorous but intuitive way?

Following some text books, the Lagrange multiplier theorem can be described as follows. Let $U \subset \mathbb{R}^n$ be an open set and let $f:U\rightarrow \mathbb{R}, g:U\rightarrow \mathbb{R}$ be $...
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1answer
95 views

Is Floer homology always isomorphic to the singular homology of some space?

After I studied Morse homology, I'm now studying the following Floer homology theories : 1) Symplectic Floer homology ; 2) Floer homology of lagrangians ; 3) Heegard-Floer homology ; ...
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1answer
37 views

Vector subbundle and frame field relation

Question: Let $E \to M $ be a vector bundle of rank $k$. Suppose that for each $p \in M $ we are given a subspace $E'_p$ of $E_p$ and consider the set $\displaystyle E' = \bigcup_{p \in M} E'_p $....
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34 views

An area form of a unit sphere centered at the origin; also considered a patch given by stereo graphic projection

An area form of a unit sphere centered at the origin; also considered a patch given by stereo graphic projection from the sphere deleted the north pole. OP is listed in a practice final exam of my ...
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2answers
74 views

Grassmanian $(2, 4)$ homeomorphic to $S^2 \times S^2$

Prove that the Grassmanian manifold $G(2, 4)$ of all real two-dimensional planes in $\mathbb{R}^4$ that pass through the origin is homeomorphic to the product of two two-dimensional spheres $S^2 \...
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0answers
39 views

What is area form?

Text book says: The area form $dS$ of a surface $S\subseteq \mathbb R ^3$ is defined as for any positively oriented orthonormal frame $\{E_1, E_2\}$, $dS(E_1,E_2)=1$. Then given parametrization $ x:...
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1answer
38 views

“Helocoidal ” surface of Gauss curvature -1

What (2 parameter) parametrisation holds for a surface Gauss curvature = -1 spanned between two helices: $$ ( \cos t, \sin t, t) ( -\cos t, -\sin t, t) ? $$ $ t= constant$ as a parameter does ...
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0answers
52 views

The tension field is a vector field

Recall that the tension field of a function $f:(M,g)\rightarrow (N,G)$ is given in local coordinates by \begin{align*} & \Delta_gf^k+g^{ij}\hat{\Gamma}^k_{mn}\partial_if ^m\partial_jf^n \\ \end{...
3
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2answers
110 views

The integral of a function on manifold and differential form

When we want to integrate a function f over a manifold M, we may meet some problems, for example, the problem showed in the picture below: Then people used differential form to integrate. But it ...
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3answers
2k views

Why is the Jacobian matrix the transpose of what I would think it'd be/usefully be (total derivative is a synonym) (EDIT: I was a total wally)

I'm sorry this isn't a yes/no/am-I-right question but I seriously cannot see why the Jacobian/total derivative matrix is what it is? I am also using it as LaTeX practice (for maths) hence the barely ...
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1answer
43 views

elements of $Z$ can be written uniquely as $z=p d^{n+1}x+p_A^\mu dy^A \wedge d^nx_\mu$

Let $X$ be an oriented $n+1$-dimensional manifold which coordinates on it are denoted $x^\mu$, $\mu =0,1,...,n$ and let $\pi_{XY}:Y\to X$ be a finite dimensional fiber bundle and fiber coordinates on $...
3
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1answer
466 views

Tangent space to a surface at boundary points

Let $M$ be a $2$-dimensional compact oriented surface in $\mathbb R^3$ with boundary $\partial M$. For any $p\in M \setminus \partial M$ tangent vectors are defined as speed vectors of smooth curves $\...
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1answer
64 views

A question about pullback of the K-form

Let $M$ be an oriented $m$-dimensional manifold. Suppose the support of $\omega$ is in an open subset $U$ of $M$, and $\phi \colon U \to R^m$, $\psi \colon U \to R^m$ are two different charts on $M$ ...
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0answers
19 views

Hypersurfaces of a hemisphere

Let $S_+^{n+1}$ be the open hemisphere of the standard euclidean sphere centered at the north pole and let $M^n$ be a compact, connected and oriented hypersurface of $S_+^{n+1}$. Is it true that if $M$...
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1answer
31 views

How to reparametrize a geodesic such that $\nabla_V V = 0$

In Nakahara's section on geodesics he says that $\nabla_V V = fV$ can be reparametrized to give $\nabla_V V = 0$ if we use the reparametrize the tangent vector components $t \rightarrow t'$ such ...
1
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1answer
36 views

If G is a compact semisimple Lie group, then its Killing form is negative definite

Theorem: If $G$ is a compact semisimple Lie group, then its Killing form is negative definite. In its proof: Since $G$ is compact there is an Ad-invariant inner product on $\mathfrak g$. Since each ...
2
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1answer
46 views

Lie bracket and inner product

$X, Y, N$ are vector fields on a riemannian manifold $M$. $\langle X, N \rangle(p)=0 $, $\langle Y, N \rangle(p)=0$ at some point $p$. Show $\langle [X,Y], N \rangle(p)=0$. I want to use $[X,Y]=XY-...
3
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0answers
46 views

How to define integration along surfaces in $\Bbb R^4$?

In $\Bbb R^4$ we have curves, bi-dimensional surfaces and hypersurfaces. We integrate vector fields along curves and in hypersurfaces using a normal direction. I know that the adequate object to ...
4
votes
2answers
54 views

Volume of spherical shell with $dr$ thickness

Let's consider two spheres in the $(x,y,z)$ 3D-space, both centered in the origin: the inner with radius $r$ and the outer with radius $r + dr$. To compute the volume of the spherical shell between ...
2
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1answer
73 views

Problem solution hint about boundary of boundary of chains from Arnold' book mathematical method

On his book Mathematical Methods of Classical Mechanics, (Chapter 7, Section 35, Problem 10), Arnold asks to show that the boundary of boundary of any chain is zero. He gives hint saying: by the ...
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0answers
144 views

Sard's Theorem Using Integration?

How exactly do we clean up this heuristic proof of Sard's theorem, from Schwarz' `Differential Topology for Physicists' using the Jacobian $J(f)(x)$: "A heuristic justification for Sard's theorem ...
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0answers
44 views

How can we visualize a tensor?

I would greatly appreciate it if someone could explain to me how to visualize a tensor in the analogous way that we visualize a vector. Thanks!
4
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2answers
187 views

complex submanifolds in complex euclidean space

Assume all manifolds are without boundaries. In Euclidean space $\mathbb{R}^n$, there are many submanifolds (Whitney Embedding Theorem). In complex Euclidean space $\mathbb{C}^n$, are there any ...
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20 views

Ring of smooth functions on a manifold and localization with respect to a multiplicative system

Take $X$ a smooth manifold and $x\in X$. It can be shown that the germ of smooth functions around $x$, $C^\infty(X)_x $ is equal to the algebraic $S^{-1}C^\infty (X)$ where $S$ is the set of smooth ...
5
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1answer
56 views

Classification of Möbius Transformations

We know how to classify the points on a surface,by looking the Gaussian curvature at a point in order to guess the shape of the surface near that point.On the other hand we classify the Möbius ...
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21 views

Reversible function

I need help. For which $(r, θ, φ) ∈ \mathbb{R}^3$ is the function $$f(r,\theta,\varphi)=\begin{pmatrix}x(r,\theta,\varphi)\\ y(r,\theta,\varphi)\\z(r,\theta,\varphi)\end{pmatrix}=\begin{pmatrix}r\sin ...
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0answers
25 views

the map from the horizontal bundle is a submersion or an immersion

Let $\pi \colon M \to B$ be a riemannian submersion and $g$ the metric on $M$. Then we get the vertical subspace $ \mathcal{V}_x = \ker d_x \pi$ and the horizontal subspace $\mathcal{H}_x= \mathcal{V}...
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0answers
38 views

On finding a second countable basis for the tangent bundle $TM$

Let $M$ be a manifold. I want to show that the tangent bundle $TM$ is second countable. I know that for a given chart $(U, \phi)$ on $M$ we have a homeomorphism $D_{\phi}$ between $TU$ and $\phi(U) \...
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3answers
32 views

How do I find the normal vector at point $p$ on a cylinder $x^2+y^2=1$?

How do I find the normal vector at point $p$ on a cylinder $x^2+y^2=1$? I would find this normal vector on point $p$ with any graphic of a function like $(-z_x,-z_y,1)$, but in this case I have no $z$ ...
2
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1answer
51 views

Pullback of euclidean metric on the disc.

$\newcommand{\Im}{\operatorname{Im}}\newcommand{Re}{\operatorname{Re}}$Consider the biolomorphism $$f : D \to H$$ where $H$ is the complex upper hyperplane $\{\Im(z) > 0\}$ and $D$ is the unic disc,...
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1answer
36 views

Top cohomology of a non-orientable smooth surface with boundary.

I would like to know what the singular relative cohomology $H^2(M,\partial M;\mathbb{Z})$ of a smooth connected surface with boundary $M$ is. In the orientable case I did the following: The zero-th ...
4
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1answer
69 views

Computing Curvature of a Connection (Dirac Monopole)

I'm trying to verify some computations in a paper I'm reading and am feeling a little lost. In particular I haven't been able to properly compute curvature of a connection acting on a line bundle. ...