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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34 views

Relation between differentials of perturbations of vector fields

Let $A$, $B$ be smooth submanifolds of a smooth manifold $M$ and $X\in C^\infty(TM)$ a vector field such following its flow $\xi^X$ gives a diffeomorphism between $A$ and $B$. Suppose also that ...
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0answers
35 views

Harmonic maps and angle preservation

I have the following question: What are the angle preservation properties of harmonic maps? Conformal maps preserve angles exactly, but they distort lengths. In this sense a conformal map is the ...
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3answers
539 views

Length of loxodrome

On a sphere with radius $R$, find the length of a loxodrome which starts at the equator and makes an angle $\gamma$ with all the meridians. (No equations for such a loxodrome are given, and should be ...
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1answer
72 views

Question about Definition of Boundary in Stokes' Theorem

I was wondering if what my teacher said was correct and complete in that in Stokes' Theorem the "boundary curve" of a surface can be defined as the mapping along the boundary of the two-dimensional ...
3
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1answer
85 views

Submanifold of $2\times 2$ Complex Matrices Using Transversality?

Let $M$ be the set of all $2\times 2$ complex matrices with $a_{21}=\bar a_{12}$. $M$ is a smooth manifold diffeomorphic to $\mathbb{R}^6$. Let $W$ be the subset consisting of all matrices in $M$ with ...
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1answer
100 views

Inclusion mapping in conformal compactifications

The conformal compactification of the prototype Euclidean space $\mathbb R^n$ can be carried out by embedding $\mathbb R^n$ into $\mathbb R^{n+1,1}$ with Minkowski signature. One then considers the ...
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2answers
156 views

Are there algebraic subgroups of Lie groups which happen to be Lie groups on their own but not Lie subgroups?

Let $G$ and $H$ be Lie groups and $i:H\to G$ an injective group homomorphism, not necessarily continuous. Once $i$ is continuous, it follows easily that $i$ is differentiable and even an immersion. ...
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1answer
23 views

Lift of isometries of spherical space forms

If we have an isometry between two spherical space forms, then it is said that it lifts to an isometry of the sphere. Why is that?
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1answer
43 views

Gradient of a parametric form

I want to find the gradient of a parametric form. So say you have the form $$(x,y,z) = (f(u,v),g(u,v),h(u,v))$$ and now I want to find and the gradient in parametric form. How do I do that? The ...
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1answer
33 views

Extending a Homotopy from $X \times [0,1]$ to $X \times \mathbb{R}$

In Jeffery M. Lee's Manifolds and Differential Geometry Exercise 1.77: For smooth manifolds $X$ and $Y$, show that if $f_0: X \rightarrow Y$ and $f_1: X \rightarrow Y$ are $C^r$ homotopic then ...
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1answer
60 views

What does it mean for the group of rotation matrices to have a “manifold structure”

I have just been exposed to rotation matrices, and it is showing extremely strong connection with group theory and differential geometry which I am both totally inept at. Can someone in simple term ...
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1answer
38 views

Wedge product (Differential Geometry) [closed]

How can we show $$\alpha_p\wedge\beta _q=(-1)^{pq}\beta_q\wedge\alpha_p$$ for the wedge product of a p- and q- form
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1answer
79 views

Inductively prove that $L_{\mathbb{X}}i_\mathbb{Y}=i_{[\mathbb{X},\mathbb{Y}]}+i_\mathbb{Y}L_{\mathbb{X}} $

Let $\mathbb{X}$, $\mathbb{Y}$ be vector fields on $U \subset \mathbb{R}^n$. Prove that $$L_{\mathbb{X}}i_\mathbb{Y}=i_{[\mathbb{X},\mathbb{Y}]}+i_\mathbb{Y}L_{\mathbb{X}} $$ using induction. Assume ...
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1answer
73 views

Is every hypersurface in $\mathbb{R}^n$ the boundary of an open domain?

We know if $\Omega \subset \mathbb{R}^{n}$ is a bounded $C^k$ domain, then its boundary $\partial\Omega$ is a $C^k$ compact hypersurface of dimension $n-1$. Is it true that every $m-$dimensional ...
3
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1answer
67 views

Volume of a paracompact manifold

It is stated, without proof, in Wald (1984) (General Relativity) that given any connected manifold $M$ (which is by definition paracompact), one may define a volume measure $\mu$ such that $\mu[M]$ is ...
4
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1answer
44 views

Bundle metric and connection on trivial vector bundle

I read this: Let $(M,g)$ be a compact Riemannian manifold and let $W$ be a vector bundle (rank $n$) over $M$ with $h_W$ a bundle metric of $W$ and $D$ a bundle connection of $W$. I choose $W$ ...
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1answer
43 views

Why $U$ generates $G$ as Lie group?

In line 2 of the proof, why is their intersection non-empty?
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0answers
51 views

Integral in Gauß Bonnet theorem

I just read a text about the Gauß Bonnet theorem. If I have a function $f:\Omega \rightarrow M$ defining a two-dimensional manifold $M$ with a boundary that is parametrized by a curve $c: I ...
3
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1answer
45 views

Variation on Stokes Theorem for Manifolds (2)

Let $\omega \in \Omega^0(\mathbb{R}^{2}\setminus\{0\})$ be a $0$-form such that $d\omega=0$. Is the following statement true: For any compact, oriented, $0$-dimensional submanifold $M$ of ...
8
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1answer
48 views

constant-curvature Riemannian metric for Bring's surface

There is a well-known and very symmetric space that is called either "Bring's curve" or "Bring's surface", depending upon the context. (Bring was a Swedish mathematician in the 18th century.) Let's ...
3
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1answer
86 views

Finding the flow of a pushforward of vector field

Let $\mathbb{X}$ be the vector field on $\mathbb{R}^2$ given by $$ \mathbb{X}(x,y) = (y,x). $$ Compute the flow $\Phi_t$ of $\mathbb{X}$. Let $F:\mathbb{R}^2\to \mathbb{R}^2$ be the ...
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0answers
52 views

Non-commuting flows and obtaining a new expression about the pullback of a function

Let $U \subset \mathbb{R}^n$ and be an open set. If $\mathbb{X}, \mathbb{Y} \in \mathcal{X}(U)$, the set of smooth, complete vector fields on $U$. Let $\Phi_t,\Psi_s$ are their respective flows and ...
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0answers
24 views

Find a complex-valued $g(u,v)$ such that $L_\mathbb{Y}g=img$

Let $F$ be a diffeomorphism between open $U$ and $V$ in $\mathbb{R}^n$. Let $\mathbb{Y}$ be a vector field on $V$ and $\omega$ a $k$-form on $V$. Given the identity ...
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0answers
53 views

Show that $\Psi_{t*}\mathbb{X}-\Phi_{t*}\mathbb{X}=\mathbb{X}-\mathbb{Y} $

Let $\mathbb{X},\mathbb{Y}$ be vector fields and let $\Phi_t$ denote the flow of $\mathbb{X}$. Given that $\displaystyle \frac{\partial}{\partial t}\Phi_{t*}\mathbb{Y} ...
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1answer
62 views

Why klein Bottle is 4-D?

I am wondering that Klein Bottle is 4-D. Can any body tell me how it is possible? I can give coordinates for each point of the Klein Bottle with 3 values. Then how it can be 4-D? What is immersion? ...
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0answers
40 views

Show that $\int_b d\omega=0$ where $b(s,t)=\Phi_s(c(t)) $

Let $c:[0,1]^k \rightarrow \mathbb{R}^n$; $t \mapsto c(t)$ be k-cell with $k < n$. Let $\mathbb{Y}$ denote a vector field on $\mathbb{R}^n$ with flow $\Psi_s$. Define a $(k+1)$-cell $b:[0,1]^{k+1} ...
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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
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1answer
37 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
563 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 ...
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5answers
92 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 ...
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0answers
67 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
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1answer
75 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
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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 ...
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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
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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 ...
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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 ...
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2answers
57 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
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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 ...
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0answers
48 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 $\{ ...
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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 ...
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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 ...
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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 ...
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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
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2answers
348 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. ...
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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 ...
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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 ...
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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
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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 ...