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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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?
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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 ...
4
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1answer
355 views

Product of manifolds & orientability

I'm studying orientability of manifolds currently and I'm having trouble to prove the following: $M\times N$ is orientable iff $M$ and $N$ are orientable. I am able to prove that the product is ...
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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?
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2answers
95 views

$f:\mathbb{R}^n \to \mathbb{R}$ has expansion $\sum_i g_i(x)x^i$

Problem 2-35 on page 34 of Spivak's Calculus on Manifolds states If $f: \mathbb{R}^n: \to \mathbb{R}$ is differentiable and $f(0) =0$, prove that there exist $g_i: \mathbb{R}^n \to \mathbb{R}$ ...
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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 ...
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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 ...
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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.
5
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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 ...
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2answers
128 views

do Carmo Differential Geometry Exercise 4.4.20 [closed]

Let $T$ be a torus of revolution which we shall assume to be parametrized by$${\bf x}(u, v) = ((r\cos u + a)\cos v, (r\cos u + a)\sin v, r \sin u).$$Prove that If a geodesic is tangent to the ...
4
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2answers
22 views

Why don't global coordinates always exist for a manifold?

Let $M$ be a manifold and $(\phi,U)$ a patch. Then $\phi(P)=\bar{x}=\begin{bmatrix} x^1\\ x^2\\ \vdots\\ x^n \end{bmatrix}$ for each $P$ in $U$. But each $P$ in $M$ is in some patch, so this ...
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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 ...
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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 ...
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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 ...
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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 ...
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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?
1
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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 ...
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0answers
52 views

Vectors in tangent space to a manifold independent of coordinates

In Nakahara's book, "Geometry, Topology and Physics" he states that it is, by construction, clear from the definition of a vector as a differential operator $X$ acting on some function ...
4
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2answers
100 views

Translating french paper into English

I am currently studying a french paper on Einstein manifolds by Berard Bergery and I have doubts that my translation of the following sentence is correct: "De plus, puisque $G$ agit par isometries, ...
1
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1answer
31 views

Bi-asymptotic geodesics in Visibility manifolds

I'm thinking about some properties of geodesics in visibility spaces. Here I give some definitions: A Riemannian manifold $(M,g)$ with riemannian distance $d$ is said to be a visibility manifold if ...
4
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2answers
109 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 ...
1
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1answer
48 views

Open neighborhoods in the definition of a manifold

At the beginning of Spivak's "A Comprehensive Introduction to Diff. Geom." (p.3), in the definition of a (topological) manifold $M$, every point $x$ has a neighborhood $U$ that is homeomorphic to ...
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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
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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
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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
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1answer
34 views

Why is the image of a smooth embedding f: N \rightarrow M an embedded submanifold?

I'm reading An Introduction to Manifolds (Tu) and got confused on p.123 Theorem 11.13. Let me briefly explain what was done before that. The author defines an embedding between two manifolds $f: ...
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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 ...
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0answers
48 views

Identity involving Riemann tensor

I'm reading about the Ricci tensor, and I've found the following statement that is given without proof: For a point $p$ on a Riemannian manifold, and coordinate vector fields $X_{\alpha}$, ...
2
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1answer
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 ...
1
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3answers
536 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 ...
1
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1answer
71 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 ...
0
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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 ...
1
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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. ...
2
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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 ...
0
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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 ...
4
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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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votes
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
0
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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 ...
2
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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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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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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 ...