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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Definition of “a topological manifold with corners”.

How can we define a topological manifold with corners and its corners? Then, do we use "invariance of domain" to define corners, as we really need this theorem in order to define "boundaries of a ...
4
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1answer
392 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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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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3answers
139 views

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) ...
3
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1answer
123 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 ...
3
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1answer
138 views

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 ...
3
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2answers
512 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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82 views

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 ...
3
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1answer
761 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 ...
3
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1answer
489 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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1answer
73 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 ...
2
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1answer
129 views

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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0answers
671 views

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 ...
2
votes
1answer
175 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 ...
2
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1answer
649 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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2answers
248 views

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 ...
2
votes
2answers
980 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
32 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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1answer
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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vote
2answers
663 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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vote
1answer
1k views

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
254 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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1answer
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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0answers
70 views

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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1answer
49 views

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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0answers
32 views

Lower boundary for hessian eigen values

This question appears on a differential geometry test i am trying to solve. Let $f$ be a function of $x,y$ and $k>0$ a positive real number. the function obeys the following: $f(0,0)=0$ and ...
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votes
1answer
170 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 ...
0
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1answer
110 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 $\Omega^{n}(S^{n})=\Omega^{n}_{+}(S^{n})\oplus \Omega^{n}_{-}(S^{n})$, ...
0
votes
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
66 views

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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1answer
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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 ...
71
votes
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 ...
59
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1answer
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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 ...
29
votes
3answers
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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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1answer
2k views

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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votes
2answers
695 views

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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3answers
1k views

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 ...
7
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2answers
2k views

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 ...
19
votes
3answers
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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 ...
15
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2answers
1k views

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 ...
18
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1answer
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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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4answers
1k views

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, ...
12
votes
1answer
661 views

Visualizing Commutator of Two Vector Fields

I'm reading a book on calculus, the part about vector fields on manifolds. It's a nice book, but with a severe drawback --- it has no pictures. I like how vectors are treated algebraically, as ...
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1answer
854 views

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 ...
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1answer
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Existence of a local geodesic frame

Let $(M,g)$ be a Riemannian manifold of dimension $n$ with Riemannian connection $\nabla,$ and let $p \in M.$ Show that there exists a neighborhood $U \subset M$ of $p$ and $n$ (smooth) vector fields ...
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votes
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Line bundles of the circle

Up to isomorphism, I think there exist only two line bundles of the circle: the trivial bundle (diffeomorphic to a cylinder) and a bundle that looks like to a Möbius band. Although it seems obvious ...
4
votes
0answers
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Tensors as mutlilinear maps

I am aware that many books on differential geometry define tensors as multilinear maps. Namely $$ V\otimes W := L_2(V^*\times W^*,\Bbb F) $$ I am also aware that this space is isomorphic to the tensor ...
21
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2answers
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Geometric intuition behind the Lie bracket of vector fields

I understand the definition of the Lie bracket and I know how to compute it in local coordinates. But is there a way to "guess" what is the Lie bracket of two vector fields ? What is the geometric ...
18
votes
1answer
3k views

precise official definition of a cell complex and CW-complex

I would be very grateful If someone could state a precise definition (direct one and inductive one) of a cell complex and CW-complex, since my intuition is telling that some restriction is missing and ...
12
votes
4answers
2k views

Topology of the tangent bundle of a smooth manifold

I am having trouble to understand what topology is given to the tangent bundle of a smooth manifold that allows it to be a smooth manifold itself. In my understanding, among other things the topology ...