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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Understanding twisted differential forms

I'm trying to understand twisted differential forms. I do know that they are like regular differential forms but under coordinate transformations they pick up an extra factor of the sign of the ...
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572 views

Motivation behind the definition of tangent vectors

I've been reading the book Gauge, Fields, Knots and Gravity by Baez. A tangent vector at $p \in M$ is defined as function $V$ from $C^{\infty}(M) $ to $\mathbb R$ satisfying the following properties: ...
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473 views

Surface integral for a scalar function defined on a discrete surface

Imagine a polyhedral, discrete surface embedded in $\mathbb{R}^3$. Its faces are all triangles. For each vertex, one can compute the discrete mean and Gaussian curvatures and evaluate the sum of ...
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1answer
30 views

Lie derivative of vielbein along Killing vector

We know that a vector $X$ is killing if the Lie derivative of the metric along $X$ vanishes: $\mathcal{L}_X (g_{mn})=0$ We also know that the metric can be written in terms of the vielbeins: $g_{mn}...
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Generalisation of the Poincaré Lemma

Let $\Omega \subset \mathbb{R}^3$ be an open but not simply connected domain and let $v \: \colon \Omega \to \mathbb{R}^3$ be a continuously differentiable vector field. Assume that $\textrm{curl} \, ...
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57 views

Abstract algebraic definition of dual tangent spaces

I know that if $(M,\mathcal{A})$ is a smooth manifold, the dual tangent space at $p\in M$ can be defined as $$ T^*_pM=I_p/I_p^2, $$ where $I_p$ is the ideal of the ring $C^\infty(M)$ consisting of ...
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1answer
91 views

Differentiation under the integral sign for volumes in higher dimensions

Consider a smooth convex/compact domain $D\subset \mathbb{R}^n$ and a smooth, concave function $F:D\to \mathbb{R}$. Then we can define the function that simply takes the volume of the upper contour ...
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97 views

Is a bijective smooth function a diffeomorphism almost everywhere?

Suppose I have $f: M \rightarrow N \in C^{\infty}$ a smooth bijection between $n$-dimensional smooth manifolds. Does it have to be a diffeomorphism except for a set of measure 0? I think the proof ...
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1answer
31 views

Property of the covariant derivative

I am learning to use the covariant derivative. In particular, I am trying to verify the expression $${\nabla}_{b}[(\nabla^a S) (\nabla_a S)] = 2 \nabla^a S \nabla_b \nabla_a S$$ for an arbitrary ...
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1answer
30 views

Baricenter of a region bounded by a parametric curve

I just want to ask if there exists a general rule to get the baricenter of a region bounded by a parametric curve?
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1answer
25 views

Divergent Curves and Complete Manifolds

I'm working on a problem in do Carmo's Riemannian geometry book (chapter 7, problem 5). He states that a divergent curve on a noncompact Riemannian manifold $M$ is a curve $\alpha: [0, \infty) \to M$ ...
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1answer
23 views

Preimage of principal bundle under equivariant map

Let $M$ be a manifold, $G,H$ be some Lie groups, $\sigma:G\to H$ be a Lie group homomorphism, $K\subset H$ a maximal compact subgroup of $H$ and $\tilde K:=\sigma^{-1}(K)\subset G$ . Let further $\...
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33 views

What does it mean for a tangent bundle to be parallel?

I am looking at page 8 of the paper here, just below definition 2.13. I have never come across a phrase such as '$T\mathfrak{C}\subset{TM}$ is parallel' - what does it mean for a tangent bundle to be ...
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1answer
24 views

Sequence of closed sets (Milnor's proof)

I've got a question about a descending sequence of closed sets. Milnor writes in his book "FROM THE DIFFERENTIABLE VIEWPOINT". In his proof of Sards Theorem he wrote: Let $f:U\rightarrow \mathbb{R}^p$ ...
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2answers
36 views

When are embeddings into Euclidean space unique up to ambient isometry?

Suppose I have a Riemannian smooth manifold $M$ and a smooth isometric embedding $M \hookrightarrow \mathbb{R}^n$. Is this embedding necessarily unique up to some isometry of $\mathbb{R}^n$? If not, ...
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2answers
210 views

Which concepts in Differential Geometry can NOT be represented using Geometric Algebra?

1. It is not clear to me that linear duals, and not just Hodge duals, can be represented in geometric algebra at all. See, for example, here. Can linear duals (i.e. linear functionals) be ...
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1answer
50 views

Topology, atlas, smooth manifold

Let $X$ be a set, $n\in\mathbb{N}$ and $((U_i,\phi_i))_i$ a family of subsets $U_i\subseteq X$ with injective functions $\phi_i: U_i\to\mathbb{R}^n$, which hold the following conditions: $\...
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10answers
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Shortest path on a sphere

I'm quite a newbie in differential geometry. Calculus is not my cup of tea ; but I find geometrical proofs really beautiful. So I'm looking for a simple - by simple I mean with almost no calculus - ...
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3answers
85 views

Differential Geometry for General Relativity

I'm going to start self-studying General Relativity from Sean Caroll's Spacetime and Geometry: An Introduction to General Relativity. I'd like to have a textbook on Differential Geometry/Calculus on ...
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44 views

nef Line bundles over Kähler manifolds

I am trying to understand a particular property of the first Chern class of a nef line bundle over a Kähler manifold. We know in general, let $X$ be a complete complex projective variety, and $L$ a ...
3
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2answers
158 views

Homogeneous Isotropic Riemannian Manifolds

In John Lee's book Riemannian Manifolds on page 33, Lee writes "Clearly a homogeneous Riemannian manifold that is isotropic at one point is isotropic at every point". It seems that he means that he ...
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1answer
26 views

Particle motion and frenet frame

I am given that $\hat{t}=\dfrac{\hat{x}+y'\hat{y}}{\sqrt{1+y'^2}}$ and $\hat{n}=\dfrac{y'\hat{x}-\hat{y}}{\sqrt{1+y'^2}}$ are the tangential and normal vector in frenet frame. We are considering only ...
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24 views

Restriction of a differential form vs pullback on submanifold $S^1\subset \Bbb R^2$

I was working through an exercise in Tu's book on differential geometry, (ex. 19.5) and I'm trying to get the most out of the exercise, and test my understanding, given that I've already computed the ...
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20 views

Confusion on Gaussian curvature computation

Exercise I'm attempting to find the Gaussian curvature of the catenoid $M$ parametrized by $$ f(u,v)=(a\cosh v\cos u,a\cosh v\sin u,a v). $$ I've run through the typical computations of $E,F,G,e,f,g$ ...
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22 views

parallelizable sphere product closed disk

From Wall's Surgery on Compact Manifolds, P9: Observe that $S^r \times D^{m−r}$ is parallelisable. If $m > r$, this is true, because spheres can be embedded in Euclidean space of one ...
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35 views

Why doesn't this argument show the Möbius bundle is trivial?

I wrote the following argument to prove that $S^1$ is parallelizable, that is, to show that the tangent bundle is trivial. It looks fairly reasonable to me. Let $\tau=2\pi$. We define a map $\...
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40 views

Existence of the lift of a curve

Let $(M, g)$ be a complete Riemannian manifold and let $(\tilde{M}, \tilde{g})$ its universal cover. Let $\pi : \tilde{M} \to M$ be the covering map. Let $\gamma : I \to (M, g)$ be a smooth curve ...
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34 views

Relation between nonorientability of the Möbius strip and the Möbius bundle

There are two ways in which the open Möbius strip $M$ is related to orientability: $M$ is nonorientable as a manifold; $M$ is the total space of the nonorientable line bundle $M \to S^1$. Is there ...
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2answers
37 views

What is the difference between intrinsic and extrinsic curvature?

In general relativity, energy bends spacetime. However, this doesn't mean that a fifth dimension for spacetime to "bend into" exists." That is, spacetime isn't embedded in a higher dimensional space, ...
2
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1answer
249 views

What is the torsion and curvature in modern language?

I think it can not be the torsion and curvature in the connection context, for these two are anti-symmetric in two variables, so that they must vanish on a one-dimensional space (tangent space of a ...
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1answer
114 views

first Chern class of E is first Chern class of det E

Let $\pi:E\to M$ be a vector bundle, and $\nabla$ a connection. My definition of the first Chern class is $$c_1(E)=\left[tr\left(\frac{i}{2\pi}F^\nabla\right)\right],$$ where $F^\nabla$ is the ...
3
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1answer
30 views

Normal sections

Let $S$ be a surface and let $p \in S$ with normal $n_{p}$. Let $q \in S$ nearby $p$, say within the injectivity radius of the exponential map at $p$. Consider the plane $\Pi = \operatorname{span}\{...
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1answer
25 views

Example of nonuniqueness of asymptotes of a ray

Let $(M, g)$ be a complete Riemannian manifold and let $\gamma : [0, \infty) \to \mathbb{R}$ be a ray, i.e. a unit speed geodesic such that for every $s, t \ge 0$ : $$ dist\big(\gamma(s), \gamma(t)\...
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Is $\mathfrak{su}_2 \simeq \mathbb{R}^3 \simeq \textrm{Im}\mathbb{H}? $

From what I've heard we have the following identifications: $\mathfrak{su}_2 \simeq \mathbb{R}^3$: $\left(x_1, x_2,x_3\right) \in \mathbb{R}^3 \leftrightarrow -\frac{i}{2}\begin{pmatrix} -x_3 &...
3
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1answer
27 views

Line in product mainifold

Let $(M_1, g_1)$ and $(M_2, g_2)$ be two complete Riemannian manifolds and consider the product $(M, g) = (M_1 \times M_2, g_1 + g_2)$. Let $\gamma : \mathbb{R} \to (M,g )$ be a line. I can write $t ...
2
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1answer
28 views

When to use coordinate charts to restrict a differential form

I've been trying to understand differential forms but still have some parts of confusion. In particular, it is not clear to me when to use charts to restrict a differential form and when not. For ...
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45 views

Preimage of a regular curve

My reference book for definitions of regular curve and regular surface is Do Carmo's book on differential geometry. Let $C$ be regular plane curve. Let $f:\mathbb{R}^3\longrightarrow\mathbb{R}^2$ be ...
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For a simple closed plane curve, is it possible to find a line satisfying following?

I am currently reading do Carmo's Differential Geometry of Curves and Surfaces, and I got stuck on page 33. Here, the author tries to prove that for a postively oriented simple closed curve $\alpha(t)$...
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0answers
50 views

computational insight behind why connections fix the shape of surface

Based on a video lecture, I had some queries. If we just have a manifold [M-set,O-topology,A-atlas] say $S^2$, this manifold represents a football or a potato equally. But once we choose a connection $...
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2answers
74 views

Exponential of Lie Groups.

When the exponential map defined a bijection between the group G and their Lie algebra? The only example I know is the Heisenberg group.
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1answer
79 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)\...
4
votes
1answer
466 views

Question about Definition of Boundary in Stokes' Theorem

I was wondering if what my teacher said was correct and complete in that in Stokes' Theorem the "boundary curve" of a surface can be defined as the mapping along the boundary of the two-dimensional ...
3
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0answers
59 views

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

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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27 views

Vector bundle over an open set of $\mathbb{R}^n$

I can't see or understand if it is true or not if all vector bundles on over an open set of $\mathbb{R}^n$ are trivial or not. Is there an easy way to see it? The problem comes from the fact that we ...
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21 views

How to know whether a contact form is only defined locally or globally?

As described e.g. here the following is the standard contact form on $\mathbb R^{2n+1}$: $$ \omega = dz + \sum_{k=1}^n x_k dy_k$$ Similarly, the following is the standard contact form on $S^{2n+1}$: ...
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1answer
22 views

Trace reverse tensor/matrix operation “carrying through” an operator

For some second rank tensor $h_{\mu\nu}$ on a Riemannian manifold with metric $g_{\mu\nu}$, one can write the trace-reverse of it as: $\bar{h}_{\mu\nu}=h_{\mu\nu}-\frac{1}{2}g_{\mu\nu}h$, where $h=h_{\...
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3answers
989 views

What are the applications of Differential Geometry in Robotics?

I am taking up a grad level course on Differential Geometry. Can any one please tell me the immediate applications of Differential Geometry in Robotics ? Thanks
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31 views

Surface of Revoution

Which generating circles ($x=constant$) appear to be geodesics? Why ? There is a hint: Imaging laying a ribbon on the surface. I could not find a way to approach this problem. I looked up and I ...
0
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1answer
19 views

$r$-jet of a smooth function and its fiber bundle.

Let $M$ be a smooth manifold of dimension $n$. Let $E$ denote the bundle of germs of smooth functions on $M$. For every stalk $E_x$ we can define the ideal $$I_x^k=\{ f \in\mathcal{C}^{\infty}(M) \...
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1answer
19 views

Explicit formulation of hermitian form and corresponding alternating form

I can't understand the following basic thing: Let $X$ be a complex manifold of dimension $n$ and $T_xX$ its tangent space in $x\in X$. Then on $T_xX$ we can define a hermitian form $\sum_{i=1}^ndz_id\...