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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I don't get the relationship between differentials, differential forms, and exterior derivatives.

I don't get the relationship between differentials, differential forms, and exterior derivatives. (Too many $d$'s getting me down!) Here are the relevant (partial) definitions from Wikipedia; ...
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0answers
20 views

Exponential map and convergence

Suppose that $M$ is smooth compact manifold and let $y \to x$. Let also $f \in C^{\infty}(M)$ be a smooth function. I consider the expression $\exp_y^{-1}(x)(f)$: then it follows that it converges to ...
3
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2answers
55 views

Differential forms in Physics

In the context of physics, I just read about the the symplectic 2-form $\omega$ on a symplectic manifold $M$ of dimension $2n$. Unfortunately, I could not follow a few arguments. I.e. it was said ...
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1answer
52 views

Is my graph a tree?

Let M be a smooth connected manifold. G is a group act on M cocompactly and suppose there is a harmonic function $h$ on M with minimal energy.$h:\rightarrow [0,1]$ such that h is nonconstant and ...
0
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1answer
42 views

How to get the result in this way?about determinant

In theory of surfaces using the fact that $\vec v \in T_p(M)$=$\alpha\vec x_u+\beta\vec x_v$ and $ S_p(\vec v)\times \vec v=\vec 0, $ deduce that a nonzero tangent vector can be a principal ...
2
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1answer
36 views

Explaining problem in Gadea's “Analysis and Algebra on Differentiable Manifolds”

I have a lot of trouble trying to explain to myself what the author did in problem 1.102 (the answer is in the link): Let $TM$ be the tangent bundle over a differentiable manifold $M$. Let ...
3
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1answer
50 views

How to show the following vector bundles are equivalent?

Given a smooth sub-manifold $X$ of $\mathbb{R^n}$ and define the diagonal in $X \times X$ to be $$\triangle = \{(x,x) \mid x \in X \} \subset \mathbb{R^n}\times \mathbb{R^n}$$ and normal bundle to ...
3
votes
1answer
31 views

Defining functions for connected sets

Let $\Omega \subset \mathbb{R}^n$ an open, bounded and connected set with a $C^2$ boundary and a function $\rho \in C^2(\mathbb{R}^n)$ such that $$ \Omega = \{ x \in \mathbb{R}^n : \rho(x) < 0 ...
7
votes
2answers
83 views

Top degree de Rham cohomology determines an orientation

Let $M^n$ be a smooth, compact, orientable, connected manifold. We know then that $H^n_{dR}(M^n)\simeq \mathbb{R}$ by the map $[\omega]\mapsto \int_{M^n} \omega$. I was wondering if, given an ...
5
votes
2answers
55 views

Curvature tensors and bivectors

At the beginning of the paper "The curvature of 4-dimensional Einstein spaces", by Singer and Thorpe, the authors define the space $\mathcal{R}$ of curvature tensors of the vector space $V$ as the set ...
1
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1answer
30 views

Equation of an osculating plane to a curve (2 variables)

Maybe I am just reading this wrong, but here goes: Find the equation of the osculating plane to the curve \begin{cases} x = a\cos(\theta) \\ y = a\sin(\theta) \\ z = a(1+\sin(\theta)) ...
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1answer
32 views

Calculate the pushforward of smooth map between manifolds

Let $\Phi : GL(n)\rightarrow Sym(n)$ be defiened by $\Phi (A)=AA^T$. I can not see how to get the "right" pushforward, I.e I want help in understanding the pushforward $\Phi _*:M_I(n)\rightarrow ...
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1answer
30 views

Codifferential and corresponding homology theory

This is the kind of a natural question which can come to mind after completing the standard course in differential geometry and homology theory: lety us start with a smooth manifold $M$. One can ...
3
votes
1answer
27 views

The set of all critical points of a smooth map is closed

Let $f : \mathbb{R}^m \to \mathbb{R}^n$ be a smooth map. How do I show that the set of all critical points of $f$ is closed in $\mathbb{R}^m$? (Here, a critical point is a point $x \in \mathbb{R}^m$ ...
0
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1answer
37 views

What is meant by “trace on any pair of indices”?

I am reading the book Riemannian Manifolds, written by John M. Lee. On page $53$, the author defines a connection on the tensor bundle $\text{T}_l^k(M)$ and says it satisfies (d) $\nabla$ commutes ...
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0answers
50 views

How to use chain rule on it?

Let $M$ be a surface in $\mathbb{R}^3$ oriented by $\vec n$. For a fixed number $\varepsilon\neq0$, let $F:M\to\mathbb{R}^3$ be a mapping such that $F(p)=p+\varepsilon\vec n(p)$ and denote ...
2
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1answer
77 views

Riemann surfaces with Riemann Roch theorem, linear fiber over an elliptic curve

Let $g:\mathbb{C}\times \mathbb{C^*}\rightarrow \mathbb{C}\times\mathbb{C^*}$ defined by $g(z,w)=(w^n z,\alpha z)$ where $0<|\alpha|<1$. Let $G$ be the cyclic group spanned by $g$ and $A$ the ...
2
votes
3answers
75 views

What does it mean that we can diagonalize the metric tensor

On a Riemannian manifold $M$, the matrix representation is diagonalisable, cause the tensor is symmetric. What is the physical meaning behind this? I mean, in Riemannian geometry, we always get a ...
105
votes
5answers
3k views

Why can't differentiability be generalized as nicely as continuity?

The question: Can we define differentiable functions between (some class of) sets, "without $\Bbb R$"* so that it Reduces to the traditional definition when desired? Has the same use in at least ...
4
votes
2answers
377 views

How to compute curvature tensors for general n-dimensions?

I keep coming across calculations like this, Consider a metric on an $n+2$ dimensional manifold given as, $ds^2 = 2dudr + 2L(u,r)du^2 -r^2d\Omega_n^2$ Then apparently once can write down the Ricci ...
2
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0answers
19 views

Orientation at the boundary for manifold with corners: the simplex

Consider the $n$-simplex $$\Delta[n]:=\{(t_{1},\dots,t_{n})\in \mathbb{R}^{n}\: : \: 0\leq t_{1}\leq t_{2}\leq \dots \leq t_{n}\leq 1\}.$$ This is a manifold with corners. The cofaces map $d^{i}\: : ...
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2answers
60 views

Diffeomorphism ( differential geometry) [closed]

What is the Geometrical interpretation of diffeomorphism in context of differential geometry ?
6
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1answer
66 views

When vectors act on scalars.

Background. I've been struggling through an introduction to differential geometry this semester. Recently, a tiny part of what we've been learning "clicked" for me, and to solidify this, I'd like ...
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0answers
36 views

How to redifined the parametrization on surface?

Let $M$ be a surface in $\mathbb{R}^3$ oriented by $\vec n$. For a fixed number $\varepsilon\neq0$, let $F:M\to\mathbb{R}^3$ be a mapping such that $F(p)=p+\varepsilon\vec n(p)$ and denote ...
6
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2answers
69 views

Shouldn't these two definitions for curvature agree?

In $\mathbb R^n$ the defintion of curvature of a smooth regular curve $\gamma : \mathbb R \to \mathbb R^n$ is $$ \kappa (t) = \|\gamma''(t)\| / \|\gamma '(t)\|$$ In $\mathbb R^2$ the definition for ...
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1answer
65 views

Boundary of the boundary of a manifold with corners

A point of a manifold with corners is a boundary point by definition if one of its coordinates is $0$ by some (hence in all) chart with corners (see here). In the same page one can read: The ...
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0answers
18 views

equivalence relation between two basis in a vector space

having the same orientation is an equivalence relation, for the different basis in a vector space. What is the purpose of equivalence relation in the sentence above?
0
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1answer
42 views

Easy example of unit speed plane curve?

I was trying to find a non-trivial example of a unit speed plane curve. The reason is I want something to work with but if I start with a non-unit speed curve and then do the arc length ...
4
votes
1answer
55 views

Calculation of $A'(0)$ (first variation of the area functional).

I'm trying to do the calculation that shows that a surface in $\Bbb R^3$ is area minimizing if and only if the mean curvature is zero. I'm getting a sign wrong and I'm going crazy, I need help. ...
5
votes
1answer
37 views

G-P Exercise, immersion except at origin, what does its image look like?

(This is not a duplicate of another question on math.stackexchange, as that other question just basically asks for the answer to the question below, of which I have provided an answer to. My question ...
2
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0answers
33 views

Flat Riemannian manifold

Is it true that a Riemannian manifold is flat, if and only if a coordinate transformation $f$ exists, such that the geodesics after transformation is in linear form ...
3
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1answer
37 views

Find a surface that has positive constant curvature that is not open subset of sphere

Can some one find a surface that has positive constant curvature that is not open subset of sphere. I know every connected and compact surface with positive constant curvature is sphere. I need ...
2
votes
1answer
41 views

Tangent space change of bases

Let $M\subset \Bbb{R}^m$ be a $k$-dimensional differentiable submanifold. Let $(\varphi, U)$ and $(\psi, V)$ be two charts for $p\in M$ with $\varphi(x)=p$ and $\psi(y)=p$. Then we have two bases for ...
5
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0answers
29 views

When a given family of curves are geodesics of some affine connection?

Let $M$ be a two-dimensional manifold and let $\mathcal C$ be a family of smooth paths on $M$. How to understand whether this family is actually a family of (possibly reparametrized) geodesics of some ...
2
votes
1answer
27 views

integrability condition on a surface

Let $v_1$ and $v_2$ be two vector fields on a two-dimensional surface. If I have a PDE on the surface \begin{align*} \nabla_{v_1} f(x) &= A(f(x), x)\\ \nabla_{v_2} f(x) &= B(f(x), x) ...
0
votes
0answers
23 views

When are geodesically generated surfaces everywhere spacelike?

Suppose that $\langle M, g\rangle$ is a Lorentzian manifold, and that $\xi$ is a timelike vector in $T_pM$, at some point $p \in M$. Let $S$ be a surface consisting of points that lie on some ...
8
votes
1answer
561 views

Is the “Constant Rank Theorem” the same as the “Domain Straightening Theorem”? Which theorem is which?

Wikipedia says that the inverse function theorem is a special case of the "constant rank theorem". I'm pretty sure this is supposed to be the same theorem as the "Rank Theorem" on p. 47 of Boothby ...
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0answers
27 views

Glueing smooth functions give a smooth function if reparametrized

Given $\mathbf{r}:(t_0-\varepsilon, t_0+\varepsilon)\to\mathbb{R}^2$ be a $C^{1}$ application, with $\mathbf{r}'(t)=(0,0)\Longleftrightarrow\ t=t_0$, and $$\mathbf{r}(t)=\begin{cases} \mathbf{r}_1 ...
2
votes
1answer
31 views

Finding the evolute of a parabola

I previously tried to find the evolute of a parabola by using parameterisation by arc length. It didn't work. While I was hoping for an answer I kept working on it and came up with the following ...
4
votes
1answer
72 views

Momentum a cotangent vector?

Imagine we have a particle described by $x \in M$, where $M$ is some manifold, then it is very intuitive I think that a velocity is an element of the tangent space at $x$, so $x' \in T_{x}M.$ Thus, by ...
3
votes
1answer
54 views

Geometric intuition about the exterior derivative

Let $M$ be a smooth manifold. One $k$-form is a section of the bundle $\bigwedge^k T^\ast M$, that is, if $p\in M$ and $\omega$ is a $k$-form then $\omega_p$ is one $k$-linear alternating real ...
8
votes
1answer
69 views

What makes differential forms special

There are two natural structures defined on differential manifolds, namely tangent bundle and its dual cotangent bundle. An element of tangent bundle $TM$ at a base point $p\in M$ can be described by ...
2
votes
0answers
24 views

Proof of Reeb's theorem without using Morse Lemma

I'm trying to prove Reeb's theorem as stated in Milnor's Morse Theory. That is, suppose we have an $n$-manifold $M$ together with a smooth function $f$ with exactly two critical points (both ...
5
votes
2answers
55 views

under what conditions can orthogonal vector fields make curvilinear coordinate system?

I am considering n-dimensional Euclidean space $\mathbb{R}^n$. For any $x\in\mathbb{R}^n$, $v_1(x), \cdots, v_n(x)$ are orthogonal vectors. As functions of $x$, $v_i$'s are differentiable and non-zero ...
2
votes
2answers
1k views

Geodesics which are lines of curvature to surfaces in Euclidean space

How to show that if a curve C in a surface is both a line of curvature and a geodesic, then C is a plane curve. Thanks
6
votes
1answer
38 views

First exercise of Guillemin-Pollack. [closed]

If $k < l$ we can consider $\mathbb{R}^k$ to be the subset $\{(a_1, \dots, a_k, 0, \dots, 0)\}$ in $\mathbb{R}^l$. Show that smooth functions on $\mathbb{R}^k$, considered as a subset of ...
0
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2answers
39 views

How to find the tangent plane to a given point on a surface?

How can you find the tangent plane to a given point on a surface? (Verbal descriptions preferred) I'm thinking you can find the "vector versions" of two directional derivatives (maybe the partial ...
1
vote
1answer
27 views

Finding evolute of parabola

I was trying to solve the following exercise when I got stuck: Find the centres of the osculating circles of the parabola $(t,t^2)$. My idea was to first reparameterise with respect to arc lenght ...
4
votes
1answer
89 views

Module of differentials in the functorial approach to schemes and quasi-coherent modules

Recall that for a functor $X : \mathsf{CAlg}(R) \to \mathsf{Set}$ from commutative $R$-algebras to sets one can define quasi-coherent $\mathcal{O}_X$-modules as "compatible" families of $A$-modules ...
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1answer
37 views

How to calculate the differential of a local chart?

I've this problem. If I have a manifold $M$ of dimension $m$ and a local chart $\varphi:M\rightarrow \varphi(M)$ which is the matrix that represents the differential of this application? Thanks in ...