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 ...

learn more… | top users | synonyms (1)

3
votes
0answers
81 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
votes
1answer
752 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
votes
1answer
483 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 $E_X$ ...
2
votes
1answer
69 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
votes
1answer
124 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 ...
2
votes
0answers
646 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
172 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
votes
1answer
613 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 ...
2
votes
2answers
247 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
965 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 ...
1
vote
1answer
27 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 ...
1
vote
2answers
660 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 $r = c + d\cos(3α)$ ...
1
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 ...
1
vote
1answer
246 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 ...
1
vote
1answer
276 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 ...
0
votes
0answers
69 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 ...
0
votes
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 ...
0
votes
0answers
31 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 ...
0
votes
1answer
155 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
votes
1answer
109 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 ...
0
votes
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 ...
21
votes
1answer
2k views

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 ...
70
votes
2answers
2k views

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
votes
1answer
2k views

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
1k views

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 ...
15
votes
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. ...
18
votes
2answers
651 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 ...
17
votes
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 ...
15
votes
2answers
997 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 ...
7
votes
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
1k views

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 ...
10
votes
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, ...
17
votes
1answer
3k views

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
12
votes
1answer
652 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 ...
12
votes
1answer
833 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 ...
9
votes
1answer
1k views

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 ...
7
votes
4answers
456 views

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
386 views

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
votes
2answers
1k views

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 ...
17
votes
1answer
2k 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
1k 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 ...
8
votes
2answers
394 views

Trying to understand the use of the “word” pullback/pushforward.

Essentially, my question is the following : Is everything we call "pullback" or "pushforward" an actual categorical pullback/pushout? I have seen tons of pullbacks in differential geometry but we ...
8
votes
1answer
254 views

Equivalence of Definitions of Principal $G$-bundle

I've finally gotten around to learning about principal $G$-bundles. In the literature, I've encountered (more than) four different definitions. Since I'm still a beginner, it's unclear to me whether ...
7
votes
3answers
326 views

Why is the Riemann curvature tensor the technical expression of curvature?

According to my textbook on general relativity (Sean Carrol's book) and differential geometry, the Reimann curvature tensor is the technical expression of curvature. What makes the tensor so special? ...
4
votes
1answer
364 views

Reference for Gauss-Manin connection

I wish to understand the notion of ``Gass-Manin connection''. I have some understanding of differential geometry, topology and algebraic geometry. Where should I begin? IF the sources are freely ...
20
votes
3answers
281 views

Is an isometric embedding of a disk determined by the boundary?

Suppose we cut a disk out of a flat piece of paper and then manipulate it in three dimensions (folding, bending, etc.) Can we determine where the paper is from the position of the boundary circle? ...
13
votes
1answer
862 views

Research in differential geometry

I am an 3rd year undergrad interested in mathematics and theoretical physics. I have been reading some classical differential geometry books and I want to pursue this subject further. I have three ...
11
votes
3answers
801 views

What is the motivation for differential forms?

I am that point in my mathematical career where I am learning differential forms. I am reading from M.Spivak's Calculus on Manifolds. So far I have gone over the tensor and wedge products and their ...
9
votes
3answers
953 views

Texts on Principal Bundles, Characteristic Classes, Intro to 4-manifolds / Gauge Theory

I am looking for a textbook that might serve as an introduction to principal bundles, curvature forms and characteristic classes, and perhaps towards 4-manifolds and gauge theory. Currently, the only ...