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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Help finding a good book on Finsler geometry

I want to learn more about Finsler geometry. I have just studying the book "An Introduction to Riemann-Finsler Geometry" by Bao, Chern and Shen, but i would like to study Finsler Geometry approach to ...
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2answers
88 views

Finding the critical points of a quadratic form restricted to projective plane

I have a quadratic form $f(x) = x^t A x$ where A is 3x3 real symmetric and $f$ satisfies $f(x) = f(-x)$ and now restricted to $||x|| = 1$ this is a well defined map on the projective plane (when ...
2
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73 views

Orthornormal basis and Dual basis

If $e_a$ is an orthonormal basis for vectors and $\theta^a$ the dual basis for coordinate vectors. How to prove that metric is expressed as $ds^2=\delta_{ab} \theta^a \theta^b$ and $e_{a}^{i}\...
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66 views

Rotation by $90°$ in differential geometry

Let $f: \Omega \subset \mathbb{R}^2 \rightarrow \mathbb{R}^3$ be a parametrized surface and $\nabla_{c'}c'$ be the covariant derivative of a curve $c:I \rightarrow \Omega$ that is parametrized by arc-...
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69 views

Does every differentiable ruled surfaces possess a global ruled parametrization?

According to my notes, a differentiable ruled surface of $\mathbb R ^3$ is a 2-dimensional $C^k$ submanifold of $\mathbb R ^3$ that can be described as a union of straight lines. I'm working on some ...
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66 views

How do I flow in a circle on a set of vector fields?

Consider a set of complete vector fields on a manifold. They each have an associated one parameter group of diffeomorphisms related to the generated flow. What is a necessary and sufficient ...
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1answer
98 views

For a surface with $K=0$ everywhere, show that the holonomy group reduces to the identity element.

Consider a connected surface $S$ embedded in $\Bbb R^3$ and let $\alpha$ be a closed path in $S$ connecting $p\in S$ back to itself. Now we define $P_{\alpha}$ as the effect of parallel transport on a ...
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66 views

We can get the geodesic only by the first geodesic equation in hyperbolic plane?

The two geodesic equations are: $$\frac d{ds} (Eu' + Fv') = 1/2(E_u {u'}^2 + 2F_u u'v' + G_u {v'}^2) $$ $$\frac d{ds} (Fu' + Gv') = 1/2(Ev{u'}^2 + 2F_v u'v' + G_v{v'}^2) $$ And in the hyperbolic ...
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1answer
61 views

last step in proof of existence of coordinate vector field

This is problem 7 on page 172 of Spivak's Differential Geometry pt. 1. Given a smooth manifold $M$ and a smooth vector field $X$ on $M$, Check that if the coordinate system $x$ is $x = \chi^{-1}$ ...
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1answer
52 views

Help with the definition of a manifold with boundary

In the definition of a manifold with boundary, we often work with the closed upper half-space $H^{n}=\left\{ x\in\mathbb{R}^{n}\,|\, x^{n}\geq0\right\}$ , and endow $H^{n}$ with the subspace ...
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1answer
20 views

Is $(-\infty,0)\times S$ for a compact closed manifold $S$ a “manifold with boundary and cylindrical ends”?

I read the following definition from this paper. Definition: Let $N$ be a Riemannian manifold with boundary $\partial N$. We say $N$ is a manifold with boundary and cylindrical ends if there ...
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1answer
40 views

Confused over k-chains and their boundaries.

I am writing a short report on de Rham cohomology, and I'm approaching it from a geometric perspective, much like (and with reference to) this article (written by a MSX member) http://www3.nd.edu/~...
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1answer
46 views

Is the smooth map with constant rank dense?

Let $M$ and $N$ be two Riemannian manifold. Assume that $f:M\rightarrow N$ is a smooth map and $\dim M < \dim N$. Can we find a smooth map $g$ $C^{k}$-close to $f$ which is immersive at each point ...
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53 views

Map projection for the surface $z=xy$ onto $\mathbb{R}^2$

(Hope I used the right tag; please correct if not.) Clarification: In the title, I mean "map projection" in the geographical sense of projecting the surface of a sphere onto a plane, but instead of ...
4
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1answer
287 views

Parametrised vs Regular Surfaces

Two types of surfaces in $\mathbb{R}^3$ are usually studied in introductory books on differential geometry: Parametrised or immersed surface: Is an immersion $F:U\rightarrow\mathbb{R}^3$ from an ...
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65 views

3D Involute of constant geodesic/ Gauss curvatures

Is my following proposition correct? Unwinding a taut surface-contacting geodesic thread to trace an involute with constant geodesic curvature $ k_{g}$ is possible only on $ \mathbb R^2 $ ...
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1answer
82 views

What does being diffeomorphic mean in the context of configuration spaces?

A sphere space can serve as a "model space" for any configuration space that is diffeomorphic to the sphere space. This is a quote from my text book (Principles of Robot Motion: Theory, Algorithms,...
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1answer
88 views

Characterisation of local affine diffeomorphisms

I've got a question about local affine diffeomorphisms between affine manifolds. There ist a good characterisation about affine diffeomorphisms of connected affine Mannifolds: Let $f,g\colon (M,\...
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1answer
57 views

Non-constant rank of a smooth map and orthnormal basis in the normal bundle

Assume $M,N$ are two Riemannian manifolds and $f: M\rightarrow N$ is a smooth map. Suppose $dim M =m < dim N =n$. Let $\Sigma$ be the graph of $f$, that is, $\Sigma =(x, f(x))$ for $x\in M$. My ...
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0answers
71 views

Local expression of a differential form

During the course, we have defined differential forms as maps $$ \omega : D(M) \times \cdots \times D(M) \rightarrow \mathcal{C}^\infty(M), $$ where $D(M)$ are the global derivates of the manifold $M$....
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1answer
70 views

Derivative first fundamental form

Let $X,Y: I \rightarrow T_{\gamma}\Omega$ be vector fields along a curve $\gamma: I \rightarrow \Omega \subset\mathbb{R}^2.$ Now, in our lecture it was claimed that the derivative $\frac{d}{dt} g(X,Y)...
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1answer
44 views

Unifying Perspectives on Discrete/Continuous Differentiation

I was prompted by some recent readings, and also by this question, to try to rectify the fact that my notions of discrete and continuous differentiation have slipped away from one another. I'm most ...
4
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0answers
85 views

Co-Area formula in Riemannian geometry

I wonder if the following holds true: Let $z:[0,1]\times B^{n-1}_r(0)\to(M^n,g), (t,p)\mapsto z(t,p)$ be a diffeomorphism a.e. onto its image with respect to the Lebesgue measure on $A:=[0,1]\times B^{...
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1answer
238 views

Spivak Calculus on Manifolds, Theorem 5-2

In the proof Theorem 5-2 of Spivak Calculus on Mannifolds how is \begin{align*} V_2\cap M=\{f(a):(a,0)\in V_1\}? \end{align*} (That $\{f(a):(a,0)\in V_1\}=\{g(a,0):(a,0)\in V_1\}$ is clear.) Edit: ...
3
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4answers
312 views

Tangent vectors: arrows vs. derivatives

I have a very hard time accepting the differential-geometric definition of a vector as a derivative operator, $$v = v^{\mu} \partial_{\mu}.$$ I want to make sure that the following line of reasoning ...
2
votes
2answers
90 views

Equivalent definitions of a surface

do Carmo Differential Geometry of Curves and Surfaces defines a regular surface as per the below post. Lee Introduction to Smooth Manifolds defines an embedded or regular surface to be an embedded or ...
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52 views

Limits of visualizing $p$-forms?

On page 90 of Gravitation, Misner Thorne and Wheeler state the following: Stacks of surfaces, individually or intersecting to make "honeycombs", "egg crates", and other structures ("differential ...
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1answer
45 views

Metric properties

Let $f: \Omega \rightarrow \mathbb{R}^3$ be a submanifold in $\mathbb{R}^3$ and also $f' : \Omega' \rightarrow \mathbb{R}^3$ another one. Now if $f(\Omega) \cap f' ( \Omega')$ is a regular curve $c: I ...
2
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3answers
238 views

Divergence theorem in complex analysis

I am revisiting my understanding of integration by parts in several complex variables, but I have run into an apparent contradiction. This shows my understanding is flawed, which is somewhat ...
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1answer
55 views

Gauss-Bonnet Theorem - Notation

Why is the equality in red true? $\bf{e}^{'}$ and $\bf{e}^{''}$ form an orthonormal basis of the tangentplane w.r.t $\gamma(s)$, which is unit speed.
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Does the classifying map of a fibre bundle only depend on the transition functions?

Does the classifying map of a fibre bundle only depend on the transition functions? Precisely, Let $\xi$ and $\eta$ be two fibre bundles over $B$, whose transition functions are same, both of their ...
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1answer
106 views

Is it possible for a manifold to have a normal vector that is zero everywhere, if so, would this indicate that the manifold is non-orientable?

Basically I've been thinking about defining a non-orientable three-dimensional metric space via defining the normal vector and looking to see if there is two possible vectors for the same point. I'm ...
0
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1answer
201 views

effect on first fundamental form of a surface by isometry

Show that applying an isometry of $\mathbb{R}^3$ to a surface does not change its first fundamental form. What is the effect of a dilation ? This is a problem from Presley book, and it has a ...
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1answer
76 views

Deriving generators for $H^1(T)$: what are $dx$ and $dy$?

By trial and error I found that $dx,dy$ are generators of $H^1_{dR}$ of $T=S^1\times S^1$. Verifying that they generate the first cohomology group is not difficult. My problem is: I found them by ...
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1answer
196 views

What can we say about a vector bundle with trivial unit sphere bundle?

If you want to avoid the back story to this question, feel free to skip the paragraphs between the horizontal lines. Yesterday Georges Elencwajg asked me the following question (I'm paraphrasing): ...
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1answer
55 views

Generators of $H^1 (T)$: take two

Previously, I asked about how to prove that $dx + dy$ is a generator of the de Rham cohomology group of the torus. Now it occurred to me that $dx$ and $dy$ are both also generators of $H^1(T)$. ...
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2answers
110 views

Relation between volume form and cross product

Euclidean three-dimensional space (it's simpler). Defining $\eta={e^*}^1 \wedge {e^*}^2 \wedge {e^*}^3$, with $\{{e^*}^1,{e^*}^2,{e^*}^3\}$ dual of the orthonormal basis, and indicating the classic ...
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3answers
53 views

The image of the map $\alpha: [-1,1] \to \mathbb{R}^2, t \mapsto (t^3,t^2)$ is not a submanifold of $\mathbb{R}^2$

Let $M$ be a smooth manifold of dimension $m$ and $N \subseteq M$. Then $N$ is said to be a submanifold of $M$ of dimension $n$ ($\leq m$) if for every point $p \in N$ there is a chart $(U,\phi)$ ...
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0answers
50 views

Space generated by a reflection

Suppose I embed a mirror (not necessarily plane) in some space (say a manifold). Is there a theory that tells you how to classify the "space" generated by the reflection (the one you see if you were ...
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0answers
35 views

Poincare inequality on balls to general open subset

Let $M$ be an n-dim compact Riemannian manifold with $Ric \geqslant -(n-1)$, it's well known that the following Poincare inequality holds for any function $f\in W^{1,2}(M)$ $$ (\int_B |f-\frac{1}{m(B)}...
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1answer
78 views

diferential equation system differential operators method

$x'-3x+2y=t$ $y'+2x=e^t$ it is asked to solve by the mentioned method $\Delta(D)=D(D-5)$ $\Delta_1=1-e^t$ $\Delta_2=-2t-2e^t$ $yD^3(D-5)(D-1)=0$ $xD^2(D-5)(D-1)=0$ When solving for the ...
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1answer
91 views

What is the definition of a properly-immersed arc?

I am going over a paper in which a theorem includes a condition that certain arcs are "properly-immersed geodesic arcs" in a surface $\Sigma$ with nonempty boundary. I have done a search, but I have ...
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150 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 $$h=\alpha^...
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0answers
362 views

Exponential map is surjective for compact connected Lie group

How do I show that for every compact connected group $G$, the exponential map $\exp \colon\mathfrak{g} \rightarrow G$ is surjective? I tried to find the proof on the internet but most of them are ...
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1answer
90 views

Is the tangent-cotangent isomorphism orientation preserving?

Consider $(M,g)$ a Riemannian manifold. Let's define $\varphi : TM\rightarrow T^{\ast}M$ by $\varphi(p,v):=(p,g(v,.))$, for $p\in M$ and $v\in T_{p}M$. Here, $TM$ stands for the tangent bundle and $T^...
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1answer
44 views

What's wrong in this prop about volume form if we drop “oriented”?

I was studying Prop 15.29 from Lee's Introduction to Smooth Manifold and I asked myself what's wrong with this proof if we drop the oriented assumption. I know that I'd came up with a non zero $n$-...
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1answer
212 views

Proving a nowhere vanishing vector field on 2D manifold implies $TU\cong M\times S^1$

So, I am trying to solve the following problem. Suppose you have a nowhere zero smooth vector field on a 2 dimensional oriented compact manifold. Prove that the unit tangent bundle $TU$ is ...
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1answer
112 views

Topological covering + local diffeomorphism gives smooth covering

I got stuck at some point while working on this part of an exercise from Lee's Introduction to Smooth Manifolds, 2nd edition. The part which I am stuck on is to prove (one of the directions of ...
3
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0answers
36 views

global function defining a $\mathcal{C}^1$ submanifold

Let $M$ be a $\mathcal{C}^1$ submanifold of dimension $1$ of $\mathbb{R}^2$. Then for each $x\in M$, there is a neighbourhood $U$ of $x$ and a $\mathcal{C}^1$ function $f:U\to \mathbb{R}$ such that $M\...
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1answer
214 views

Covariant derivative and geodesic

Let $f: U \subset \mathbb{R}^2 \rightarrow \mathbb{R}^3$ be a surface patch. Then if we have two vector fields $$X = \sum_i \xi^i \frac{\partial f}{\partial u^i}$$ and $$Y = \sum_i \eta^i \frac{\...