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)

1
vote
2answers
44 views

Question concerning tensors

As some of you may have seen from my previous question(s), I am working through Spivak's Calc on Manifolds and this happens to be the first time I've been introduced to tensors formally, though I have ...
6
votes
1answer
72 views

Finding two inequivalent closed, non-exact $1$-forms on $T = S^1 \times S^1$

I've been studying the torus and the first cohomology group $H^1_{dR}(T)$ for a couple of weeks now. I finally had a breakthrough of understanding and would like to kindly request the community to ...
3
votes
0answers
41 views

When is a linear map of 1-forms a pullback?

Every diffeomorphism $\phi: M\to N$ between two-dimensional compact oriented Riemannian manifolds induces a linear map on one-forms $L:\Omega^1(M)\to\Omega^1(N)$ given by the pullback of $\phi^{-1}$. ...
2
votes
1answer
30 views

What sort of algebraic structure describes the “tensor algebra” of tensors of mixed variance in differential geometry?

differential geometer in training here. With regards to my background, I learned differential and Riemannian geometry from O'Neill and Lee's series. I'm working on my algebra background (which is ...
0
votes
3answers
35 views

Is saying each pt of a topo. manifold has nbhd homeomorphic to R^n the same thing as saying there is a local coordinate system at each point?

Is saying each point of a topological manifold has a neighborhood homeomorphic to $\Bbb{R}^{n}$ the same thing as saying there is a local coordinate system at each point? I'm not really sure what ...
3
votes
2answers
49 views

Find a parametrization of the intersection curve between two surfaces in $\mathbb{R^3}$ $x^2+y^2+z^2=1$ and $x^2+y^2=x$.

Find a parametrization of the intersection curve between two surfaces in $\mathbb{R}^3$ $$x^2+y^2+z^2=1$$ and $$x^2+y^2=x.$$ I know that $x^2+y^2+z^2=1$ is a sphere and that $x^2+y^2=x$ is a circular ...
0
votes
1answer
62 views

Gauß and mean curvature

I was wondering whether the Gauß, mean curvature and shape operator of a surface actually depend on the chosen parametrization? Under a reparametrization of $f: \Omega \subset \mathbb{R}^2 ...
0
votes
1answer
72 views
+50

Isometric and conformal map

We defined conformal and isometric maps for surfaces $f,g: \Omega \subset \mathbb{R}^2 \rightarrow S \subset \mathbb{R}^3$. Under a reparametrization of $f$ I understand a diffeomorphism $\Phi : M ...
1
vote
0answers
24 views

Zero Gauss curvature surfaces spanning a loop

EDIT 1: Spanning across a given arbitrary closed boundary/loop a surface can be defined with zero mean curvature H in 3-space as minimal surfaces. Likewise can a surface be defined with zero Gauss ...
4
votes
2answers
88 views
+50

Hessian of a function on Riemannian manifolds

Let $(M,g,\nabla)$ be a Riemannian manifold with metric $g$ and Riemannian connection $\nabla$. The hessian of a function $f:M\to R$ is defined by: $$H^f(X,Y)=g(\nabla_X\ \ \operatorname{grad} ...
0
votes
1answer
14 views

vertical/horizontal asymptotes - general understanding

Do vertical asymptote only exists in fractions? My taught was yes. Can the curve cross a vertical asymptote? My taught was no. Can the curve cross a horizontal asymptote? my taught was yes. Thanks
1
vote
2answers
30 views

Embed curves in the plane

The strongest version of Whitney's embedding theorem says that every smooth real $n$-dimensional manifold $M^n$ (Hausdorff and second-countable) can be embedded in $\mathbb{R}^{2n}$. This should mean ...
0
votes
1answer
13 views

Must a Developable Surface be Tangent Developable or a Generalised Cone/Cylinder?

I've commonly seen that tangent developable surfaces, Generalised cones and generalised cylinders are developable surfaces. (see http://en.wikipedia.org/wiki/Developable_surface) But are these the ...
2
votes
0answers
59 views

Multivariable Calculus or Differential Geometry (Analysis on Manifolds) after single variable calculus

Background: Applied Mathematics program, finished with single variable calculus, and in parallel with basic analysis. (Not knowledge of multivariable calculus yet) Please feel free to recommend ...
4
votes
2answers
97 views

How to evaluate this integral: $\oint dx$?

I am trying to understand differential forms. Now I tried to evaluate $$ \oint_{S^1}dx$$ I should get anything non-zero but I don't know how to do it (even though I know the result). If $S^1$ in ...
0
votes
0answers
22 views

Use Frenet Frame and Pythagorean Theorem.

Suppose we have a curve $c(t)$ where $t$ goes from $a$ to $b$. $c$ has positive curvature and a frenet frame(TNB). Choose $\rho > 0$ and small. and define: $f(t) = c(t) +\rho B(t)$ $g(t) = c(t) ...
1
vote
0answers
44 views

Basic Differential geometry: Shortest path between two points in R^3 is straight.

Given two points P and Q in $\mathbb{R^3}$, we want to show that the shortest distance between them is through a straight line. let $c(a) = P$ and $c(b) = Q$ and $c(t)\neq P$ for $t>a$(One ...
2
votes
1answer
49 views

Relationship between differential forms and coordinates

For the purpose of this question let us restrict our considerations to smooth $3$-manifolds. So the manifold $M$ we consider here is endowed with smooth coordinate charts $(x,y,z)$. What I have ...
3
votes
1answer
40 views

Embedded Submanifolds Have a Unique Smooth Structure

Let $M$ be a smooth manifold. An embedded submanifold of $M$ is a subset $S$ of $M$ such that $S$ is a topological manifold under the subspace topology induced by $M$, endowed with a smooth structure ...
3
votes
2answers
43 views

Is there a better way to show the intrinsic curvature of a cylinder is zero?

I am new to differential geometry and Riemannian geometry. I have on two separate occasions (separated by 6 months) encountered exercises where I feel like I am not giving a complete answer. ...
0
votes
1answer
29 views

Sum of Killing vector fields is a Killing vector field

Let $(M,g)$ be a Riemannian manifold. A smooth vector field $X$ is called a Killing vector field if the flow of $X$ acts by isometries, or, equivalently, if $L_X g = 0$. Now why is the sum of Killing ...
2
votes
1answer
46 views

Laplace's equation in rectangle geometry

Consider Laplace's equation in a rectangle with length and width of a and b respectively, with following boundary conditions: All the boundaries with $x < a/2$ have Drichlet boundary condition ...
3
votes
1answer
25 views

How many isometries of the (unit) 2-sphere are there?

I had a homework problem that exhibited two Killing vector fields for the 2-sphere, asked me to find a third and then asked me if there are any more. I answered no because the Lie Algebra of the ...
4
votes
2answers
37 views

Compute a parallel transport

Let $\mathbb{S}^{2} \subset \mathbb{R}^{3}$ be the $2$-sphere ($\mathbb{S}^{2} = \left\{ (x,y,z) \in \mathbb{R}^3, \; x^2+y^2+z^2 = 1 \right\}$). Let $p \in \mathbb{S}^{2}$ and $\xi \in T_{p}S^{2} = ...
1
vote
1answer
23 views

How to prove easily that geodesic is auto parallel?

I only have the elementary concept of geodesic and differential equn of geodesic. Intrinsic derivative =0 implies parallel displacement along a curve. What does it mean by auto parallel? How to prove ...
6
votes
0answers
56 views

Differentiation on Manifolds Basics

I'm having some real trouble comprehending integral curves and Lie derivatives on a Manifold. I will write out my understanding and ask the questions below. For a vector field $X$ on smooth manifold ...
0
votes
0answers
22 views

Are there two kinds of Christoffel symbols?

I am struggling to understand Christoffel symbols. Part of my confusion is that there are two kinds. So I mix up which properties belong to each and end up learning about neither. Can someone define ...
2
votes
2answers
113 views

The space $x^3-y^2=0$

Consider $\{(x,y)\in\mathbf{R}^2 \ | \ x^3-y^2=0\}$ as a subspace of $\mathbf{R}^2$. Intuitvely I understand that this is not supposed to be a differentiable manifold because it has a cusp at $0$. But ...
0
votes
0answers
25 views

Transversality of leaves to the spheres .

Consider a form in the complex plane such that its linear part is $\omega_0=\lambda_1xdy-\lambda_2ydx$ in the Poincare domain: $\lambda_1\lambda_2 \ne 0$ and $\lambda_1/\lambda_2 \notin \mathbb{R}^-$. ...
0
votes
0answers
32 views

Geometrical definition of the first fundamental form?

I am looking for a geometrical interpretation of the first fundamental form. I would guess it is something like a small line element that sits in the surface which then integrated over will give the ...
0
votes
1answer
34 views

Proof, that helical surface is a submainfold [closed]

I have to proof, that helical surface $M:= \left|\begin{array}{ccc}s\cos(t)\\s \sin (t)\\t\end{array}\right|$ s,t$\in R$ is 2 dimensional submanifold. How to do it?
0
votes
1answer
20 views

Curvature of a parallel surface

I have found a couple of questions that deal with the basic concepts, I am asking about, but nothing that is quite the same as my question. So .... This is a question from an MIT OpenCourseWare ...
1
vote
1answer
29 views

Discrete Gauß and geodesic curvature

Imagine that you have an n-polygon $S$ and you wanted to calculated the discrete Gaussian or gedoesic curvature. How are they defined? If $p$ is a vertex of $S$ then Gauß-Bonnet suggests that the ...
0
votes
0answers
46 views

Vanishing Christoffels symbols

Under what conditions does there exist a parametrization of a surface, for which the Christoffel symbols are zero. I heard that has something to do with "flat connection". I would like to see proofs.
0
votes
1answer
29 views

What is a comoving basis?

I have read that the tangent vector, principal normal vector and binormal vector consistute a comoving orthogonal basis. But in this context what does comoving mean?
5
votes
0answers
63 views

Examples of categorical adjunctions in analysis and differential geometry?

In a lot of introductory texts on category theory, it seems like the majority of examples come from algebraic topology, algebra, and logic. Are there any good examples of adjunctions in analysis and ...
0
votes
1answer
36 views

Connected components of Lorentz Group $O_1(3)$

Let us consider the set of all vector isometries of the space $\mathbb{E}^3_1$, $O(1,3)$. I know this group has four connected components but I can't prove it. Could someone help me? I'm completely ...
0
votes
1answer
33 views

Topology on the tensor Bundle $T^{r, s}(M)$?

Let $M$ be a smooth manifold and for $r, s\geq 0$ define the tensor bundle: $$T^{r, s}(M):=\bigcup_{p\in M} T_pM.$$ I'm trying to understand its topology. I'm following Homology and Curvature written ...
-1
votes
0answers
98 views

Hate calculus, but want to learn differential geometry? [closed]

Title. I really, really, really hate calculus. I do find the techniques beautiful, but I find the computations absolutely dreadful. I'm also intrigued by differential equations, but once again, the ...
5
votes
2answers
70 views

Why are contact structures studied from a cohomological, rather than homological, perspective?

As far as I know, contact structures are studied from a "cohomology/dual" perspective, meaning mostly from the perspective of contact forms and their respective kernels, instead of from a ...
0
votes
1answer
37 views

$(2n-1)$-form is closed [closed]

Consider in $\mathbb{R}^{2n}$ differential forms$$\omega = dx_1 \wedge dx_2 \wedge \dots \wedge dx_n\text{ and }\theta = \sum_{n+1}^{2n} (-1)^{j-1}x_jdx_{n+1} \wedge \dots \wedge dx_{j-1} \wedge ...
-3
votes
2answers
51 views

If $f : M\rightarrow N$ be immersion then $f_*$, derivative of $f$ is an immersion. [closed]

Suppose that $f : M\rightarrow N$ be immersion. Prove that $f_*$, derivative of $f$, is immersion too?
0
votes
0answers
34 views

Computation of the first fundamental form of ruled surfaces

It is possible to prove that ruled surfaces can be parametrized as follows: $\overrightarrow{X}(t,u)=\overrightarrow{\beta}(t)+u \overrightarrow{w}(t)$ where $\|\overrightarrow{w}(t)\|^{2}=1$ and ...
1
vote
1answer
40 views

Isometries are affine transformations

I want to show that, if $(M,\mathrm{g})$ is a Riemannian manifold, $\nabla$ is the covariant derivative from the Levi-Civita connection, and $f:M\to M$ is an isometry, then ...
1
vote
1answer
45 views

A question regarding Jacobi fields and families of geodesics

I'm trying to show that for any one-parameter family of geodesics $\gamma(s,t)$ (where $\gamma(s_0,t)$ is a geodesic for any constant $s_0 \in (-\epsilon, \epsilon)$) defined on a Riemannian manifold ...
4
votes
2answers
110 views

Elementary proof of the fact that any orientable 3-manifold is parallelizable

A parallelizable manifold $M$ is a smooth manifold such that there exist smooth vector fields $V_1,...,V_n$ where $n$ is the dimension of $M$, such that at any point $p\in M$, the tangent vectors ...
3
votes
2answers
50 views

Why do a set of continuous transformations form a manifold?

I am reading Sean Caroll's book on GR, and he defines manifolds to be "a space that may be curved and have a complicated topology, but in local regions looks just like R$^n$. Here by "looks like" we ...
2
votes
2answers
49 views

Poincare disk and Poincare half plane

My book claims that the Möbius transforms are isometries of the Poincare half plane model. Thus, the metric is preserved under these maps. But I know that the Poincare disk can be derived from ...
2
votes
0answers
36 views

Laplacian on sphere with differential forms [closed]

I want to express the Laplacian on the 2-sphere in terms of differential forms. Does anybody know how this can be done? I am not so familiar with submanifolds, thus I would appreciate help very much. ...
1
vote
0answers
42 views

De Rham Cohomology of the complement of an ellipse

I'll call $A$ the complement of an ellipse in $\mathbb{R}^2$. I know that $H^0(A)=\mathbb{R}^2$ and $H^k(A)=0$ if $k \ge 3$. What about $k=1,2$? I know that $A$ has two connected components, so ...