# Tagged Questions

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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### 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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### 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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### 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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### 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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### 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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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)\... 0answers 28 views ### 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 &... 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 ... 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 ... 0answers 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 ... 0answers 32 views ### 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)... 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 ... 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. 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)\... 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 ... 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 $$I treat it as heat equation with heat source . So, no matter the heat source is very hot or not , the ... 0answers 27 views ### Vector bundle over an open set of \mathbb{R}^n I can't see or understand if it is true or not if all vector bundles on over an open set of \mathbb{R}^n are trivial or not. Is there an easy way to see it? The problem comes from the fact that we ... 0answers 21 views ### How to know whether a contact form is only defined locally or globally? As described e.g. here the following is the standard contact form on \mathbb R^{2n+1}:$$ \omega = dz + \sum_{k=1}^n x_k dy_k$$Similarly, the following is the standard contact form on S^{2n+1}: ... 1answer 22 views ### Trace reverse tensor/matrix operation “carrying through” an operator For some second rank tensor h_{\mu\nu} on a Riemannian manifold with metric g_{\mu\nu}, one can write the trace-reverse of it as: \bar{h}_{\mu\nu}=h_{\mu\nu}-\frac{1}{2}g_{\mu\nu}h, where h=h_{\... 3answers 989 views ### What are the applications of Differential Geometry in Robotics? I am taking up a grad level course on Differential Geometry. Can any one please tell me the immediate applications of Differential Geometry in Robotics ? Thanks 0answers 31 views ### Surface of Revoution Which generating circles (x=constant) appear to be geodesics? Why ? There is a hint: Imaging laying a ribbon on the surface. I could not find a way to approach this problem. I looked up and I ... 1answer 19 views ### r-jet of a smooth function and its fiber bundle. Let M be a smooth manifold of dimension n. Let E denote the bundle of germs of smooth functions on M. For every stalk E_x we can define the ideal$$I_x^k=\{ f \in\mathcal{C}^{\infty}(M) \...
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\...