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
1answer
25 views

Deduce that there is a cup product that is well-defined

I have showed that if $\alpha$ and $\beta$ are closed forms on a smooth manifold $M$, then $\alpha \wedge \beta$ also be closed. Further, if one of $\alpha$ or $\beta$ is exact, than $\alpha \wedge \...
1
vote
1answer
33 views

Smooth function from function with singularity

Having an application $f:(t_0-\varepsilon, t_0+\varepsilon)\to\mathbb{C}$ in $C^{\infty}((t_0-\varepsilon, t_0+\varepsilon))$ with $f(t)=0\Leftrightarrow\ t=t_0$ and knowing that: $\exists\ \lim\...
2
votes
1answer
51 views

Is that application smooth?

Having an application $f:(t_0-\varepsilon, t_0+\varepsilon)\backslash\{t_0\}\to\mathbb{C}$ in $C^{\infty}((t_0-\varepsilon, t_0+\varepsilon)\backslash\{t_0\})$ with $|f(t)|=1,\forall t\in (t_0-\...
2
votes
1answer
77 views

An interpretation of $ \frac{\partial^{2}}{\partial x^{2}} $.

$ \left( \dfrac{\partial}{\partial x} \right)_{p} $ is both an element of the tangent space $ {T_{p}}(M) $ and a linear functional on $ {C^{1}}(M) $, while $ (\mathrm{d}{x})_{p} $ is an element of the ...
2
votes
1answer
53 views

Pipe-fitting conditions in 3D

Let's we have smooth (continuous and infinitely differential) curve $f(x(t), y(t), z(t)) = 0$ in 3D. Now I want to build a tube of diameter $D$ around it. Questions: What are the set of conditions ...
1
vote
1answer
27 views

Conditions on $a,b,c,d$ such that $\gamma (t)$ is regular for all $t$?

I solved an exercise in my book and I was wondering if someone could look at my answer and tell me if it is correct please? The exercise is this: Let $\gamma (t) = (a \cos t + b \sin t, c \cos t + ...
3
votes
0answers
47 views

Is this a normal form for $4$-forms on manifolds?

Starting from a $2$-form $\omega$ which is nondegenerate and closed on a $2n$-dimensional manifold, it is always possible to find local coordinates $x_1,y_1,\ldots,x_n,y_n$ so that the form $\omega$ ...
1
vote
0answers
29 views

How can I get this new Gaussion curvature and mean curvature?

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 $\overline{...
1
vote
1answer
46 views

Diffeomorphism between vector bundles

I have some difficulty solving the following problem: Let $M$ be a diffentiable manifold of dimension $m$, which admits a global base of differentiable vector fields $\{X_1,\ldots,X_m\}$; this $\{X_1(...
1
vote
1answer
54 views

Degree of smooth maps are equal $\Rightarrow$ homotopic

It is an easily proven theorem that if $f,g:M\to N$ are smooth maps that are homotopic maps between compact, connected, oriented, smooth manifolds of dimension $n$, then $\deg f=\deg g$. I was ...
0
votes
0answers
26 views

Prove that orthogonal n x n matrices form a $C^1$ surface of dimension n(n-1)/2 in $\mathbb{R^n}^2$ [duplicate]

Consider the function $F:Mat_n → Sym_n$ defined by the formula $F(A) = A∗A$. $Mat_n$ denotes the vector space of n × n matrices with real entries, while $Sym_n$ denotes the vector space of symmetric $...
1
vote
0answers
39 views

Imagening the Thurston geometries

I can (more of less) imagine how it would look if space was Euclidean, spherical of hyperbolic. But there are 8 Thurston geometries see https://en.wikipedia.org/wiki/Geometrization_conjecture how ...
7
votes
1answer
215 views

Tautological line bundle over $\mathbb{RP}^n$ isomorphic to normal bundle? Also “splitting” of transition functions

Hallo fellow mathematicians. I try to understand why the normal bundle of $\mathbb{PR}^n$ is isomorphic (in the category of vector bundles) to the tautological line Bundle. More aptly, why $\nu_{\...
3
votes
2answers
101 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 ...
13
votes
2answers
380 views

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; ...
3
votes
1answer
65 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 $\...
2
votes
0answers
40 views

Does the wedge product of bundle-valued forms induce a universal object?

Given a smooth manifold $M$ and a vector bundle $E$ over $M$, the $C^\infty(M)$-module of $E$-valued $p$-forms on $M$ is defined to be $$\Omega^p(M; E) := \Gamma_M\left( \bigwedge^p T^*M \otimes E \...
2
votes
1answer
55 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 $\...
7
votes
2answers
204 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 ...
1
vote
1answer
34 views

Globally defined exponential in a particular homogeneous space

I'm currently working in a particular conformal compactification/completion of the Minkowski space-time, but I'm stuck at showing that the exponential of every vector field in it is globally defined. ...
1
vote
1answer
133 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)) \end{...
5
votes
2answers
134 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 ...
2
votes
1answer
72 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 T_I(...
1
vote
1answer
55 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 ...
-1
votes
1answer
48 views

How to get the result in this way?about determinant [closed]

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 ...
4
votes
1answer
132 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 $x \in \mathbb{R}^m$ ...
1
vote
1answer
64 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 ...
1
vote
0answers
51 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 $\overline{...
2
votes
3answers
347 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 ...
5
votes
0answers
206 views

Second Variation of Area Functional

This is a follow up question from this one. I have proved that given a parametrized surface ${\bf x}$, the mean curvature is zero if and only if it is a critical point of the area functional. Then ...
2
votes
0answers
44 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}\: : \...
2
votes
1answer
83 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 ...
7
votes
1answer
74 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 to ...
0
votes
2answers
82 views

Diffeomorphism ( differential geometry) [closed]

What is the Geometrical interpretation of diffeomorphism in context of differential geometry ?
0
votes
1answer
174 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
94 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. ...
2
votes
0answers
46 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 $\mathbf{y}_t=\mathbf{a}t+\mathbf{y}...
5
votes
1answer
40 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 ...
7
votes
0answers
43 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 ...
3
votes
1answer
58 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
135 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 ...
6
votes
2answers
95 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 ...
1
vote
0answers
34 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 ...
1
vote
0answers
29 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 (t)...
2
votes
1answer
121 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 ...
9
votes
1answer
146 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 ...
3
votes
0answers
72 views

commutation relation of angular momentum operator in non cartesian coordinates

The angular momentum operator $J$ in quantum mechanics with the commutation relation \begin{equation*} [J_l,J_m]=i\hbar\epsilon_{lmn}J_n \end{equation*} has the structure of a Lie-algebra. It is ...
6
votes
2answers
185 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 ...
6
votes
1answer
48 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 $\mathbb{R}...
0
votes
2answers
63 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 ...