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)

0
votes
0answers
21 views

Differentiability of a function on a manifold is independent of the coordinate chart

I am currently working through Nakahara's book, "Geometry, Topology and Physics", and have reached the stage at looking at calculus on manifolds. In the book he states that "The differentiability of a ...
0
votes
1answer
39 views

Left-invariant differential form on homogeneous space

I'm trying to learn by working through an example but I got stuck with the line of reasoning provided in the text. Let $G$ be the group of matrices $$g= \begin{bmatrix} 1 &0 &\log{x} \\ y ...
5
votes
1answer
565 views

approximating geodesic distances on the sphere by euclidean distances of a transformed sphere

Is there a way to find a function $F:\mathbb S^2 \rightarrow \mathbb R^3$ of class $C^1$, minimizing $$\int_{\mathbb S^2\times\mathbb S^2}(d(F(x),F(y))−\delta(x,y))^2 dx dy$$ , where $d$ stands for ...
6
votes
5answers
93 views

Surface all of whose normals intersect at a point

I am new to differential geometry and encountered difficulty when trying to solve the following problem from Dubrovin's Modern Geometry It's the first problem in exercise 8.4: Find the surface ...
1
vote
0answers
68 views

Definition of the second fundamental form in $\Bbb L^3$, following Kühnel.

I'm having trouble understanding a few comments is Kühnel's "Differential Geometry: curves - surfaces - manifolds", in page $118$, about the definition of the second-fundamental form in ...
7
votes
1answer
76 views

Tensors as Multilinear maps?

Today I learned about Tensors as multilinear maps. I usually think of tensors as a multidimensional array of numbers with fixed transformation laws, and I am having trouble understanding how tensors ...
3
votes
1answer
46 views

The injectivity of torus in the category of abelian Lie groups

HI: I have the following question: Definition: A Lie group $T$ is called a torus if $T\cong \prod_{1\leq i\leq k} \mathbb{R}/\mathbb{Z}$ for some $k\in \mathbb{N}$. ${\bf Question}$: Is it true ...
2
votes
0answers
31 views

Differentiability of a map into $\wedge T^*M$

Let $M$ a differentiable manifold and $p \in M$. Consider $(U;x_1, \dots, x_n)$ a coordinated system around $p$. $\{dx_\phi\}_\phi$ with $\phi \in \cal{P}(\{1, \dots n\})$, $dx_\phi=dx_{i1} \wedge ...
8
votes
1answer
60 views

Does the Tangent Space Vary Continuously with The Points On a Manifold?

I recently read about Grassmannian manifolds. The following question naturally comes to mind. Let $GR_k(\mathbf R^n)$ is the grassmannian manifold of $k$ dimensional linear subspaces of $\mathbf ...
1
vote
0answers
14 views

Why do entries of Maurer-Cartan 1-form span space of left-invariant 1-forms

Suppose $G$ is a $k$-dimensional Lie group of $n\times n$ matrices of the form $[g_{ij}(x_1,...,x_k)]$ where $g_{ij}$ are smooth functions. As a follow-up to the question I posted recently, I now ...
0
votes
2answers
58 views

Variation on Stokes Theorem for Manifolds

Let $n >1$ and $\omega \in \Omega^{n-1}(\mathbb{R}^{n+1}\setminus\{0\})$ such that $d\omega = 0$. Is the following statement true: For any compact, oriented, $(n-1)$-dimensional submanifold $M$ ...
2
votes
1answer
42 views

A question about a part of the proof of the existence of the exterior derivate.

Let $M$ a differentiable manifold, $(U;x_1,\ldots,x_n)$ a coordinated system and $\omega$ a differential form which domain intersects $U$. In $U \cap \operatorname{Dom} \omega$, $\omega$ can be ...
2
votes
2answers
93 views

which surfaces have (for a large area) a constant negative curvature?

There is no surface in $ R^3 $ that can represent the complete hyperbolic plane (Hilberts theorem) so we always have to do with a surface that is not completely equivalent, has a cusp somewhere, but ...
0
votes
0answers
17 views

Assymptotic direction

Me and my classmates are interested in a visual description of an asymptotic direction at a point of a surface. The normal curvature in an assymptotic direction at a point is zero. And a curve on a ...
0
votes
0answers
49 views

Charts of $\wedge T^*M$ and $\wedge TM$.

Let $M$ a differentiable manifold and consider $\wedge T^*M$. If $(U, \phi)$ is a chart of $M$ with coordinate functions $(x_1, \dots, x_n)$, then $\{ \frac{\partial }{\partial x_i}\}_i$ and $\{ ...
0
votes
1answer
28 views

Local Homeomorphisms: Characterization

Problem Consider for simplicity a surjection $F:X\to Y$. Are these characterizations of local homeomorphisms equivalent: $$\forall x\in X:\exists U_x\in\mathcal{T}_X, V_y\in\mathcal{T}_Y:\quad ...
0
votes
1answer
29 views

What is $T^0_0(M,W)$ where $W$ is trivial vector bundle over a compact manifold $M$?

Let $W=(M \times \mathbb{R}, pr, M)$ be the trivial vector bundle over a compact manifold $M$, and define $$V=T^0_0(M,W) := T^0_0M \otimes W,$$ and $V$ is called "the vector bundle of $W$-valued ...
3
votes
0answers
35 views

Invariance of determinant of metric tensor

Given any 2-tensor on a Riemannian manifold $M$ equipped with metric $g,$ we have a coordinate-free definition of its trace: $$\operatorname{trace}(T)=g^{ij}T_{ij}= T_i^i.$$ In particular, we have ...
1
vote
1answer
19 views

Schwartz rule in differentiable manifolds.

Let $M$ a differentiable manifold and $(U,\varphi)$ a chart with coordinate functions $(x_1,...,x_n)$. Let $p \in U$. Given $f:U\longrightarrow \mathbb{R}$, $f \in C^\infty(U)$, it is possible to ...
11
votes
2answers
349 views

Checking my understanding of $T^*M$ as a symplectic manifold and the links between the classical description of Lagrangians vs this invariant way.

I am working through a book titled "An introduction to mechanics and symmetry" by Marsden and Ratiu. I have written up a brief summary trying to solidify my understanding of the general principles. ...
0
votes
2answers
36 views

Which of the following surfaces are compact and which are connected?

Which of the following surfaces are compact and which are connected? The region $z>0$ in $z=xy$ $M:x^2+y^4+z^6=1$ I think the first one is neither connected nor compact, since the surface ...
2
votes
1answer
32 views

Is the Riemannian distance function Lipschitz on a hypersurface?

Let $M$ be a compact hypersurface in $\mathbb{R}^{n+1}$ of dimenion $n$. Is it true that there exists a constant $C$ such that $$d(x,y) \leq C|x-y|$$ for all $x, y \in M$? Here $d$ is the Riemannian ...
0
votes
0answers
23 views

Time Derivative of the integral over a singular k-cube

I am stuck on this question, and was wondering if someone could provide a hint of where to start? I can't see the first step.
3
votes
0answers
125 views

Squeezed cylinder parametrization

A Cylinder is such a common surface. But is there a parametrization for an isometrically $ R^2 $ bent cylinder whose major and minor dimensions are along x, y axes? I used an approximation to ...
2
votes
1answer
99 views

Weaker definitions of Lie subgroups

A Lie group $H$ is called a Lie subgroup of a Lie Group $G$ if there is a map $i:H\to G$ which is (a) an injective immersion and (b) a group homomorphism. My questios are: What happens if we replace ...
1
vote
1answer
34 views

Constant curvature metrics on the sphere

Are there Riemannian metrics other than the standard metric induced from the euclidean space on $S^2$ such that the sectional curvature is equal to 1 everywhere? Or is this the unique Riemannian ...
1
vote
1answer
57 views

Difference between parameterization and embedding of manifolds

What is the difference between embedding and parameterization? Why, for example, we say Gauss parameterization of a convex hypersurfaces, and we don't call it an embedding?
0
votes
0answers
17 views

Rotationally non-symmetric Sine Gordon application

Has the intrinsic Sine-Gordon equation been ever used to define asymptotic lines on constant negative Gaussian curvature surfaces of Kuen, Breather or other rotationally non-symmetric surfaces ? ...
1
vote
0answers
46 views

Explicit Calculation of the Euler class for the 2-Sphere using transition functions

I have been trying to learn about characteristic classes for months now, and every time I try a simple example something goes wrong. Any insights would be greatly appreciated. I am trying to follow ...
1
vote
0answers
35 views

Asymptotic lines on $ x^2 y^2 + y^2 z^2 + z^2 x^2 = 1 $

How are asymptotic lines defined/computed on this implicit surface? EDIT1: In Monge form $ z = \frac { 1- z^2 y^2}{x^2+y^2}, $ I find later it can be done because it can at all be cast into such ...
1
vote
2answers
40 views

Give an explicit atlas for the manifold $\mathbb{R}P^3$ which is defined as the quotient of the three-sphere by the antipodal mapping.

I'd like to know if there is an explicit atlas for the manifold $\mathbb{R}P^3$ which is defined as the quotient of the three-sphere by the antipodal mapping. Thanks.
1
vote
1answer
28 views

Find an explicit atlas for this submanifold of $\mathbb{R}^4$

I'm having a hard time coming up with atlases for manifolds. I can prove using the implicit function theorem that $M = \{(x_1,x_2,x_3,x_4)\in\mathbb{R}^4:x_1^2+x_2^2=x_3^2+x_4^2=1\}$ is a ...
1
vote
1answer
29 views

Covariant derivative ambiguity

I'm studying general relativity and am running into an ambiguity with the covariant derivative. The covariant derivative acting on a scalar is, in a co-ordinate basis, simply $$\nabla_X f = X^a ...
2
votes
1answer
51 views

How does Maurer-Cartan form work

I have seen similar post asking for interpretation of the Maurer-Cartan form, but I am still struggling to understand it, so let me try to work a specific example and pose a specific question. Let ...
2
votes
2answers
103 views

Every 1- or 2-dimensional compact, connected Lie group is abelian

I know that in the 1-dimensional case the conclusion follows immediately from the fact that there exists a maximal torus (which then has to be the group itself). In the 2-dimensional case, we also ...
0
votes
0answers
43 views

Integral Curve of the vector field

If we have a 2-sphere with coordinates $x=r \cos \theta \cos \phi$, $y= r \sin \theta \sin \phi$ and $z=r \sin \theta$ and the vector field $X= (-r\sin \theta \cos\phi, r \cos \theta \sin \phi, r \cos ...
0
votes
0answers
34 views

Lie Derivative of the Jacobian?

$X,Y$ smooth manifolds. Given a map $f:X\longrightarrow Y$. Let's say for simplicity that $f$ is a diffeomorphism and defined everywhere. Let $v,w$ be smooth vectorfields on $X$ and $g\in\mathcal ...
2
votes
1answer
128 views

Problem integrating when attempting a solution with the Poincaré Lemma

d) This is part I am having troubles with. I get that $$ \hat{\mathbb{X}}_t = \left(\frac{\partial}{\partial t}\hat{\Phi}_t \right) \Phi_t^{-1} = \left(\frac{\partial}{\partial t}\hat{\Phi}_t ...
2
votes
2answers
289 views

Geodesics on torus

Describe the geodesics on Torus $$\sigma (u,v)= ((a+b \cos u)\cos v, (a+b\cos u)\sin v, b\sin u)$$ First fundamental form for torus is $$b^2 du^2 +(a+b \cos u)^2dv^2$$ Consider unit-speed ...
1
vote
0answers
36 views

The change of parameter of a regular curve is a diffeomorphism, and preserves the length

Let $C$ be a regular curve and let $\alpha:I\subset\mathbb{R}\to C$, $\beta:J\subset\mathbb{R}\to C$ be two parametrizations of $C$ in a neighborhood of $p\in\alpha(I)\cap\beta(I)=W$. Let ...
0
votes
2answers
55 views

Reparametrisation of closed not closed

I would like an example of a closed curve and a reparametrisation of the same curve that is not closed. Closed in the sense that it is periodic. i.e there exists $r\in \mathbb{R}-0$ such that ...
5
votes
1answer
53 views

The Underlying Manifolds of the Special Unitary Lie Groups SU(n)

I want to find the underlying manifolds of Lie Groups $\mbox{SU}(n)$ for general $n$. $$ \quad $$ My lecturer told us last year that the only n-spheres that admit a Lie group structure are ...
0
votes
0answers
26 views

Arc length parametrization of parameter curves of the sphere

I would like to find a parametrization of (part of) the sphere where the parameter lines are arc length parametrized. The reason is that I was asked to show that if the parameterlines of a surface ...
4
votes
1answer
87 views

A doubt about Differential Geometry Books.

I intend to read "Physics for Mathematicians" by Spivak, and he says that vols. 01 and 02 of "A Comprehensive Introduction to Differential Geometry" are necessary to understand the book. Are those ...
0
votes
0answers
32 views

Fundamental group of cusp of a negatively curved manifold

Let $M$ be a complete, noncompact Riemannian manifold with finite volume and whose sectional curvatures vary within the interval $[a,b]$, $-1\leq a<b<0$. It is known that such manifold has ends ...
3
votes
0answers
81 views

Making a gradient-like vector field a gradient vector field via choosing a Riemannian metric.

Let $\xi$ be a vector field on manifold $M^n$ which is a gradient-like vector field for a some Morse function $f$. Prove that there exists a Riemannian metric on $M$ such that $\xi$ is a gradient ...
0
votes
1answer
38 views

Locally exact vs globally exact

Why the volume form in Sphere is locally exact but not globally exact? here the integral is integral $$\int_{S^n}w$$ with $$w = \frac{1}{r} \sum_{i=1}^{n+1} (-1)^{i-1} x_i dx_1 \cdots\hat{dx_i},\cdots ...
0
votes
0answers
43 views

Riemannian Connection

How can we see for the Riemannian connection, connection 1-form with its first index lowered $\omega_{ab}=\delta_{ac}{\omega^c}_b$ is antisymmetric in a, b, i.e. $\omega_{ab}=-\omega_{ba}$. Thanks.
4
votes
1answer
193 views

Use of Poincare Lemma in solving $\nabla \times \textbf{A}(\textbf{r})=\frac{\textbf{r}}{r^3}$ UPDATED

You are given the following statement of the Poincaré Lemma: If $\Phi_t$ is a one-parameter family of diffeomorphisms on $\mathbb R^n$ (not necessarily a subgroup) and $X_t$ the vector field ...
0
votes
1answer
61 views

Sum of wedge products with one forms equals 0

Let $M$ be a smooth manifold and $n \leq dim(M)$. Let $\omega_1,...,\omega_n$ be 1-forms on $M$, such that for every $q \in M$ their evaluations $(\omega_1)_q,...,(\omega_n)_q$ at $q$ are linearly ...