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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Curvature and convexity of a plane curve

Let $\mathbf{r}:[a,b]\to\mathbb{R}^2$ be a $C^2([a,b])$ regular curve. Is it true that $\mathbf{r}$ is convex if and only if its curvature $\kappa(t)\leq 0, \forall t\in [a,b]$ or $\kappa(t)\geq 0, ...
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Shouldn't these two definitions for curvature agree?

In $\mathbb R^n$ the defintion of curvature of a smooth regular curve $\gamma : \mathbb R \to \mathbb R^n$ is $$ \kappa (t) = \|\gamma''(t)\| / \|\gamma '(t)\|$$ In $\mathbb R^2$ the definition for ...
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How to show that $\{(x,y,z)\in\mathbb{R}^3:x^4+y^4+z^4=1\}$ is diffeomorphic to the $2$-sphere.

How to show that the "squared sphere" $$\tilde{S}^2=\{(x,y,z)\in\mathbb{R}^3:x^4+y^4+z^4=1\}$$ is diffeomorphism to the standard $2$-sphere $$S^2=\{(x,y,z)\in\mathbb{R}^3:x^2+y^2+z^2=1\}?$$ ...
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why $\iint \sqrt{1+\left(\frac{\partial f}{\partial x}\right)^2+\left(\frac{\partial f}{\partial y}\right)^2}\:dx\:dy$ means surface area?

Why the following integral means the area of surface $f(x,y)=z$? $$\iint \sqrt{1+\left(\frac{\partial f}{\partial x}\right)^2+\left(\frac{\partial f}{\partial y}\right)^2}\:dx\:dy$$
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510 views

Total Derivative and Multilinear Functions

So I'm starting to work through Spivak's Calculus on Manifolds and I'm having a little trouble verifying some of the claims made in the book problems. To review: Given a function ...
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553 views

A local diffeomorphism of Euclidean space that is not a diffeomorphism

Could someone give me an example of a local diffeomorphism from $\mathbb{R}^p$ to $\mathbb{R}^p$ (function of class say $C^k$ with an invertible differential map in each point) that is not a ...
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394 views

Lebesgue measure on normal matrices

Consider the space of $n\times n$ complex matrices, and equip it with its Lebesgue measure $dX$, seen as a $2n^2$-dimensional real vector space [edit: or better, a complex vector space (see the answer ...
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63 views

Is the shortest path in flat hyperbolic space straight relative to Euclidean space?

I have the following metric $$ ds^2 = dt^2-dx^2 $$ and I wanted to prove to myself that the shortest path for this metric is straight. I used the following relation $x=f(t)$ and $$ S = ...
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Torus diffeomorphic to $S^1\times S^1$.

This is an exercise from Guillemin/Pollack's Differential Topology. In a previous exercise, I'm asked to give a complete set of parametrizations of $S^1\times S^1$, which I've succeeded in (I think) ...
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128 views

De Rham Cohomology of the tangent bundle of a manifold

I would like to compute the de Rham cohomology of the tangent bundle $TM$ of a manifold $M$. It seems to me that we can just homotopy each fibre to a point, and that this would give a homotopy ...
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If $\gamma$ is spherical, then the equation $\frac{\tau}{\kappa}=\frac{d}{ds}(\frac{\dot{\kappa}}{\tau \kappa^2})$ holds.

Question: Let $\gamma (t)$ be a unit-speed curve with $\kappa(t)\gt0$ and $\tau(t)\neq0$ for all $t$. Show that, if $\gamma$ is spherical, i.e., if it lies on the surface of a sphere, then ...
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518 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 ...
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Relation between n-tuple points on an algebaric curve and its pre-image in the normalizing Riemann surface

This question is sort of an extension to this previous question of mine, Hyperellipticity (or not!) of a Riemann surface and the singularities of the curve If one knows the multiplicity of a ...
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773 views

planar curve if and only if torsion

Again I have a question, it's about a proof, if the torsion of a curve is zero, we have that $$ B(s) = v_0,$$ a constant vector (where $B$ is the binormal), the proof ends concluding that the ...
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496 views

Tangent space to circle

I guess I am missing something obvious here. I am reading about vector bundles. (What Karoubi calls 'Quasi Vector-Bundles') An example is the sphere, where for every point $X \in S^n$ we choose $E_X$ ...
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Proof a $(2n-1)$-compact manifold

I have no idea how prove that $$\{(z_0,\ldots,z_n)\in\mathbb{C}^{n+1} \quad| \quad z_0^d+z_1^2\ldots+z_n^2=0, \quad |z_0|^2+|z_1|^2\ldots+|z_n|^2=2\}$$ is a $(2n-1)$-compact manifold. How give the ...
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75 views

Hausdorff Dimension of a manifold of dimension n?

Let's say that $M$ is a differentiable manifold of dimension $n$. (This includes that $M$ is nonempty and second countable, so that it can be embedded into some Euclidean space, and is thus ...
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finite length of a spiral

consider a "spiral" $\alpha(t)=r(t)\left(\cos(t),\sin(t)\right)$, where $r$ is $\mathcal{C}^1$ and $0\le r(t) \le 1$ for all $0 \le t$ Show that if $\alpha$ has finite length on $ [0,\infty)$ and ...
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Hard Differential Equation. Please help.

first of all I'm not a mathematician, so I apologize if any of my understanding and terminology isn't up to par. Also, I've never used this website (or any of these kind of question/answer) websites ...
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1answer
177 views

Unique Perpendicular Geodesic

Let $p < q < r < s$ be real numbers. Let $l$ be the geodesic with endpoints at $p$ and $q$ and let $m$ be the geodesic with endpoints at $r$ and $s$. (a) Prove that there is a unique geodesic ...
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684 views

Covariant derivative and surface gradient

The surface gradient of a function defined on a surface $\Gamma \subset \mathbb{R}^n$ is defined $$\nabla_{\Gamma} f = \nabla f - (\nabla f \cdot N)N$$ where $N$ is the unit normal on $\Gamma.$ How ...
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Orientation on $\mathbb{CP}^2$

I am confused by the orientation of a topological manifold. My understanding is: An orientation of a topological manifold is a choice of generator of the $H^n(M,\mathbb Z)$. So given a manifold, we ...
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996 views

Direction of the second derivative of an arclength parametrized curve

I have a question, it's so simple and stupid ._. If I have a planar curve parametrized by arc length, it's easy to show that the second derivate is orthogonal to the first derivate vector (tangent ...
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1answer
38 views

Finding evolute of parabola

I was trying to solve the following exercise when I got stuck: Find the centres of the osculating circles of the parabola $(t,t^2)$. My idea was to first reparameterise with respect to arc lenght ...
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51 views

Product Manifold: Tangent Spaces

Problem Given a product manifold. How to prove that its tangent spaces split into direct sums: $$T_{(p,q)}(M\times N)\cong T_pM\oplus T_qN$$ Attempts One could try the geometric perspective: ...
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665 views

Area of a weird ellipse shape.

A propeller has the shape shown below. The boundary of the internal hole is given by $r = a + b\cos(4α)$ where $a > b >0$. The external boundary of the propeller is given by $r = c + d\cos(3α)$ ...
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1answer
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Constant rank theorem

In Boothby's "An introduction to Differentiable Manifolds ...", page 69, Remark 4.2: I am unable to discover two cubes of same side length, as asked for. I shall feel very thankful if anyone ...
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1answer
258 views

Immersing punctured torus

I am looking for a proof (as elementary as possible) of the fact that punctured n-dimensional torus admits an immersion to $\mathbb{R}^n$. The 2-dimensional case seems to be evident, but I haven't got ...
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277 views

shortest distance between two points on $S^2$

Length of Curve in $2D$ is $l_{\gamma}(\mathbb{R}^2)=\int_{0}^{1}\sqrt{(dr/dt)^2+r^2(d\theta/dt)^2}$ Length of a curve in $3D$ is ...
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Derivative of a group action

Let $\phi : G \times M \rightarrow M$ be a group action on a smooth manifold $M$ and Lie group $G$. Then we define $$f(t):=\phi(g(t),d(t)).$$ where $g: I \rightarrow G$ and $d: I \rightarrow M$ are ...
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Vector fields that are smooth on the open unit ball $B$, are smooth on $\mathbb{R}^n$ if they are zero outside $B$

Could anyone help a little with the following problem? Fix $\varepsilon \in (0, 1)$ and choose a smooth function $h$ on $[0,\infty)$ such that $h'(t) > 0$ for all $t ≥ 0$, $h(t) = t$ for $t \in ...
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Show differential of $f:S^{m}\times S^{n}\to S^{m+n+mn}$ is injective

The problem is find to the differential of $f:S^{m}\times S^{n}\to S^{m+n+mn}$ (spheres) defined as $f(x_{0},...,x_{n},y_{1},...y_{n})=(x_{0}y_{0},x_{0}y_{1},...,x_{n}y_{1},x_{n}y_{n})$ and show it is ...
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178 views

Relation between the integral of geodesic curvature and Gaussian Curvature

I need help with an exam question: Let $S$ be a regular oriented surface such that for any simple, closed, and positively oriented curve in $S$ the value of the integral of the geodesic curvature ...
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111 views

Antipodal mapping of the sphere

Suppose we have a closed form $d\omega=0$ on $S^{n}$. If $i: S^{n} \to S^{n}$ is the antipodal map, it induces a decomposition $\Omega^{n}(S^{n})=\Omega^{n}_{+}(S^{n})\oplus \Omega^{n}_{-}(S^{n})$, ...
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1answer
74 views

what are the various fields in which circle is treated as infinite sided regular polygon?

What are the various fields in which circle is treated as infinite sided regular polygon? What I actually mean is , "can u suggest me some applications where circle is treated as infinite sided ...
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1answer
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What does it mean to say a boundary is $C^k$?

I need a explanation on what does it mean to say a boundary is $C^k$. Can anyone help me please. And also need some explanation on how to straighten boundary ?
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Can any smooth planar curve which is closed, be a base for a 3 dimensional cone?

A cone in 3 dimensions has a vertex and a base. The contour of the base is a circle which is a smooth closed planar curve. Can there be a more general cone which can have any smooth closed planar ...
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Does a Fourier transformation on a (pseudo-)Riemannian manifold make sense?

the Fourier transformation of a scalar function with respect to one variable might be defined as $\mathcal{F}\left[w\right](\omega )\equiv ...
72
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2answers
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Direct proof that the wedge product preserves integral cohomology classes?

Let $H^k(M,\mathbb R)$ be the De Rham cohomology of a manifold $M$. There is a canonical map $H^k(M;\mathbb Z) \to H^k(M;\mathbb R)$ from the integral cohomology to the cohomology with coefficients ...
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What's the largest possible volume of a taco, and how do I make one that big?

Let $f$ be a continuous, even, positive function over some interval $I=[-a,a]$ such that the total arc length of $f$ over $I$ is at least $2$, $f(0)=0$, and $f$ is increasing on $(0,a)$. View the ...
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Natural and coordinate free definition for the Riemannian volume form?

In linear algebra and differential geometry, there are various structures which we calculate with in a basis or local coordinates, but which we would like to have a meaning which is basis independent ...
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Intuitive explanation of covariant, contravariant and Lie derivatives

I would be glad if someone could explain in intuitive terms what these different derivatives are, and possibly give some practical, understandable examples of how they would produce different results. ...
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Symmetric and wedge product in algebra and differential geometry

I have been struggling with this issue for a while (and asked a similar question here), but still not found a satisfying answer. The question boils down to: which is the correct identity? $dx \, dy ...
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Geometric understanding of differential forms.

I would like to understand differential forms more intuitively. I have yet to find a book which explains how the use of the exterior product in differential forms ties into the geometrical ...
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Tensors as matrices vs. Tensors as multi-linear maps

So I read the answers in this question, and don't feel that much closer to an answer about how tensors as multi-linear maps and tensors as "multi-dimensional" matrices are truly related. For ...
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concrete examples of tangent bundles of smooth manifolds for standard spaces

I'm having trouble visualizing what the topology/atlas of a tangent bundle $TM$ looks like, for a smooth manifold $M$. I know that $$\dim(TM)=2\dim(M).$$ Do the tangent bundles of the following ...
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Motivation behind the definition of a Manifold.

A manifold $M$ of dimension n is a topological space with the following properties: a) $M$ is Hausdorff b)$M$ is locally Euclidean of dimension n c) $M$ has a countable basis of open sets. Why is ...
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recommending books for intro to diff. geometry

I was wondering if anyone could recommend some books for studying topics such as abstract manifolds, differential forms on manifolds, integration of differential forms, stoke's thm, dRham chomology, ...
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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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Car movement - differential geometry interpretation

The problem presented below is from my differential geometry course. The initial reference is Nelson, Tensor Analysis 1967. The car is modelled as follows: Denote by $C(x,y)$ the center of the ...