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)

1
vote
0answers
17 views

Pullback map distributes over wedge product (proof)

To prove that the pullback map distributes with the wedge product is it first best to prove that it distributes over the tensor product and then use the relation $$dx^{\mu_{1}}\wedge\cdots\wedge ...
2
votes
0answers
22 views

Proving that the pullback map commutes with the exterior derivative

I'm trying to prove that the pullback map $\phi^{\ast}$ induced by a map $\phi:M\rightarrow N$ commutes with the exterior derivative. Here is my attempt so far: Let $\omega\;\in\Omega^{r}(N)$ and let ...
2
votes
0answers
10 views

Homogeneous metric on a homogeneous space $G/K$ - is this the same as a $G$ - invariant metric?

I have trouble putting down the notion of a homogeneous Riemannian metric. Suppose we are given a Riemannian manifold $(M,g)$ on which a compact Lie group $G$ acts transitively by isometries (this ...
0
votes
2answers
26 views

How to prove that in a Kähler manifold without boundary $\Omega \wedge \cdots \wedge \Omega$ is closed but not exact?

Let $M$ be a compact Kähler manifold without boundary. How can I show that the volume form $$ \Omega^{m} = \Omega \wedge \cdots \wedge \Omega $$ where we have the wedge of $m$ $\Omega$s is ...
3
votes
0answers
15 views

G-P Exercise 4.8.2, proof verification.

Let $\gamma$ be a smooth closed curve in $\mathbb{R}^2 - \{0\}$ and $\omega$ any closed $1$-form on $\mathbb{R}^2 - \{0\}$. Prove that$$\oint_\gamma \omega = W(\gamma, 0) \int_{S^1} \omega,$$where ...
0
votes
3answers
34 views

What is the volume and surface area of the 1-Sphere?

I am reading a post on here that mentioned something about the 1-sphere. I know that a 2-sphere is a circle, and 3-sphere is a volume, but what is this 1-sphere and how do you calculate the volume and ...
7
votes
1answer
50 views

Questions about torsion of a curve in $\mathbb{R}^3$ and analogues of torsion in higher dimensions

Suppose we have a curve $\alpha(s) : I \to \mathbb{R}^3$ parametrized by arc-length that has nowhere-vanishing second derivative, so that we are able to define the torsion $\tau(s)$ for every $s \in ...
1
vote
1answer
21 views

Normal vector in curvilinear coordinates

Is it true that the normal vector, or, $\ddot{\mathbf r}$ always vanishes for: a helix in cylindrical coordinates a loxodrome in spherical coordinates a torus knot in toroidal coordinates When ...
3
votes
1answer
35 views

Does map induced by rotation preserve the volume form?

Let $A: \mathbb{R}^n \to \mathbb{R}^n$ be a rotation. My question is, does the map of $S^{n-1}$ onto $S^{n-1}$ induced by $A$ necessarily preserve the volume form?
16
votes
1answer
69 views

Are there spaces that 'look the same' at every point, but are not homogeneous?

A metric space is homogeneous if for any two points there is a global isometry that maps one into the other. It is locally homogeneous if any two points have isometric neighborhoods, i.e. the space ...
1
vote
0answers
26 views

How to compute the unit outer normal at the point in a curve?

Given a smooth closed curve $f(x,y)=0$, How to compute the unit outer normal at each point $(x_{0},y_{0})$ in the curve?
2
votes
1answer
41 views

Divisors of differentials.

Let $\mathbb{C}$ be the base field. Suppose $C \subset \mathbb{P}_2$ is a nonsingular projective curve of degree $d > 3$. Must it be the case that $D \in \text{Div}(C)$ is the divisor of a ...
6
votes
0answers
33 views

Angle form, 1-form, proof verification.

Check that the $1$-form $d\,\text{arg}$ in $\mathbb{R}^2 - \{0\}$ is just the form$${{-y}\over{x^2 + y^2}}\,dx + {{x}\over{x^2 + y^2}}\,dy.$$ My solution is as follows. Observe that we can ...
5
votes
1answer
62 views

Can a $1d$ space never be curved?

I was wondering about this: Wikipedia article I refer to (here I refer to the first part: metric) This wikipedia article claims that this hyperbolic space model has constant curvature $-1.$ I believe ...
4
votes
0answers
38 views

maximal linear subspaces contained in the cone over the Clifford torus.

Forgot: this is about Find a subspace of $\mathbb{R}^4$ for which $x^T*A*x$ = 0 I was a little surprised to find that, in the cone $x^2 + y^2 = z^2 + w^2$ in $\mathbb R^4,$ there are infinitely many ...
4
votes
1answer
49 views

Redundance of the Smoothness of the Inversion Map in the Definiton of a Lie Group.

$\DeclareMathOperator{\inv}{inv}$ I am trying to understand the proof of the following from this document: Let $M$ be a smooth manifold which admits a group structure such that the multiplication ...
3
votes
1answer
30 views

When are the eigenvalues of the second fundamental form equal to the principal curvatures?

I am confused about the following concerning the second fundamental form. Consider a surface $S$ $\subset R^3$ If we consider a chart at a point $p \in S$, $f$: $R^2$$\to S$ and suppose $\partial ...
1
vote
1answer
30 views

Jacobi field along every geodesic?

I stumbled over the question: Given a manifold $M$. Can there exist a vector field that is a Jacobi field along every geodesic? Now, the answer is apparently: Yes, because the $0$ vector field does ...
2
votes
0answers
23 views

Laplace-Beltrami on a Curve

Is there a way to write out Laplace-Beltrami operator explicitly for a sufficiently smooth plane curve given by implicit equation $s(x,y)=0$? I know that if we knew the parametrization of the curve, ...
1
vote
0answers
21 views

Normal coordinates and the metric tensor

I was wondering whether the metric tensor in normal coordinates can be expressed somehow in terms of the exponential map. Cause I just don't see how the metric in normal coordinates is actually ...
0
votes
0answers
6 views

show that $K=E_2[\omega_{12}(E_1)]-E_1[\omega_{12}(E_2)]-\omega_{12}(E_1)^2-\omega_{12}(E_2)^2$?

$$K=E_2[\omega_{12}(E_1)]-E1[\omega_{12}(E_2)]-\omega_{12}(E_1)^2-\omega_{12}(E_2)^2$$ where $K$ is Gaussian curvature, $E_i$'s are tangent frame field on surface $M$ in $R^3$, $v[.]$ is directional ...
2
votes
0answers
23 views

Connection and reduction of the structure group

I am writing a memoir about gauge theory. I have trouble with a small proof which should be simple and have the feeling that I am missing something obvious. I want to show that the set of connections ...
1
vote
0answers
25 views

How do I show $f_0 \sim f_1 \Rightarrow f_0 + g \sim f_1 + g$

Consider two smoothly homotopic maps $f_1,f_2:M \to S^1$ from a compact smooth $n$-manifold $M$ to the unit circle. How do I show $$f_0 \sim f_1 \Rightarrow f_0 + g \sim f_1 + g$$ for all $g:M \to ...
0
votes
0answers
7 views

anlyalytic paths through convergent cauchy sequence II

Assume we have a Cauchy sequence $\{\vec{a_i}:i\in\mathbb{N}\}$ converging to $\vec{0}$ in $\mathbb{C}^n$ such that $|\vec{a_i}|<|\vec{a_j}|$ whenever $i>j$. Can we find an analytic path ...
3
votes
4answers
75 views

Why do disks on planes grow more quickly with radius than disks on spheres?

In the book, Mr. Tompkins in Wonderland, there is written something like this: On a sphere the area within a given radius grows more slowly with the radius than on a plane. Could you explain ...
5
votes
1answer
47 views

Real Manifold … Complex Coordinates?

I'm working in an earlier edition of John Lee's book on smooth manifolds, and he has a number of problems where he represents a real manifold using complex variables. For instance in chapter 3 ...
1
vote
1answer
23 views

Cylindrically symmetric vector field

I want to prove that if $u$ is a cylindrically symmetric vector field in $\mathbb R^3$, then $$\frac{\partial u_x}{\partial x}=\frac{\partial u_y}{\partial y}$$ I've tried this by direct derivation, ...
1
vote
0answers
19 views

Isometry algebra implication from Riemannian covering

I really wish that, if $\pi:(M,\mathrm{g})\twoheadrightarrow(N,\mathrm{h})$ is a Riemannian covering, then $\mathfrak{i}(N,\mathrm{h})\leq\mathfrak{i}(M,\mathrm{g})$, where ...
2
votes
0answers
46 views

Try to show $f(x)=-x$ is an orientation preserving map from $S^n$ to itself

Consider a map $f:S^n \to S^n$ defined by $$f(x)=-x$$ and want to show that this map is orientation preserving iff $n$ is odd. What I have done is, consider the standard orientation n-form on ...
1
vote
1answer
18 views

Ruled surface out of lines of curvatures

I'm trying to proof the following statement: Let $c$ be a curve inside a surface element $f:U\rightarrow\mathbb{R}^3$ (i.e $c=f\circ\gamma$ where $\gamma:I\rightarrow U$). Then $c$ is a line of ...
1
vote
1answer
23 views

How do we find the principal unit normal to this curve?

A curve is given in cylindrical coordinates: $r=r(t)$ $\theta=\theta(t)$ $z=z(t)$ The curve is unit-speed: $(\frac{dr}{dt})^2+r^2(\frac{d\theta}{dt})^2+(\frac{dz}{dt})^2=1$ How do we find the ...
1
vote
1answer
29 views

Extension Lemma for Functions on Submanifolds

The following lemma is my question. (cf GTM218, Introduction to Smooth manifold) I can prove (b) using partion of unity as follows: $Proof$ for any $p \in S$ choose a slice chart $W_p$ centered at ...
1
vote
1answer
17 views

Relationship between euclidean metric in sphere of radius $r$ and the unit sphere.

I want to show $g_r=r^2g_1$ where $g_1$ is the (Riemannian) metric in the unit sphere induced by its inclusion in $\mathbb{R}^n$ and $g_r$ is the metric in the sphere of radius $r$ also induced by ...
0
votes
1answer
47 views

From $\mathbb{H}$ to Poincaré disc? [on hold]

What is the mapping that takes one from the Poincaré upper half plane $\mathbb{H} = \{ z\in \mathbb{C} \mid \operatorname{Im}(z)>0 \}$ to the Poincaré disc? Here $z=x+i y$.
-4
votes
0answers
30 views

Differntial Geometry [on hold]

When it comes to study unit normal,binormal & tangent vector ,it is obvious to say that all are mutually perpendicular but with the entry of acceleration vector things get out of my mind. since ...
4
votes
2answers
76 views

How do ideas in differential geometry expand upon ideas from introductory calculus

I just went through first year in mathematics and used Stewart's book for calculus. I am trying to self study differential manifold and I find many concepts such as chart, atlas very similar to that ...
0
votes
1answer
42 views

Carto-polar curve

Is there any plane curve that has the same equation in cartesian as in polar coordinates? To be more specific, is there a function such that $f(f(x)\cos x)=f(x)\sin x, \forall x$?
0
votes
1answer
23 views

Gradient of Distant Function

I am learning the Hessian comparison theorem on Riemannian manifold. It refers to the gradient of distant function. Fix $x\in M$. Let $\rho(y)=d(y,x)$, and $r:I\to M$ is a minimal geodesic curve with ...
1
vote
1answer
33 views

'Large' closed subgroup

I am working through a paper in the field of differential geometry (Yang-Mills theory) and the author writes: 'We assume the Riemannian manifold $(M,h)$ admits a large closed subgroup $K$ of the ...
5
votes
0answers
81 views

Review on Riemannian Geometry

I'm currently reading through Griffiths and Harris Principles of Algebraic Geometry, and the only subject in the foundational material section that I am not completely comfortable with is riemannian ...
1
vote
0answers
22 views

Yamabe flow, Metric times Scalar curvature?

I was watching a lecture on differential geometry on Ricci flow, when someone asked a question about "Scalar curvature being multiplied by metric" to my understanding this shall be written as ...
0
votes
1answer
33 views

Analytic paths through converging sequences in the complex space.

Assume we have a Cauchy sequence $\{\vec{a_i}:i\in\mathbb{N}\}$ converging to $\vec{0}$ in $\mathbb{C}^n$ such that $|\vec{a_i}|<|\vec{a_j}|$ whenever $i>j$. Can we find an analytic path ...
1
vote
1answer
26 views

Why is a surface of revolution injective?

Let $f:U \rightarrow \mathbb{R}$ and $g:U \rightarrow \mathbb{R}$ be smooth functions where $U \subset \mathbb{R}$ is an open set such that $f(x) > 0$ and $f'(x)^2 + g'(x)^2 = 1$ for any $x$ in ...
0
votes
0answers
12 views

angular metric in Finsler geometry

Let $(M,F)$ be a Finsler manifold. for $x\in M$, $I_{F}(x):=\{y\in T_{x}M\mid F(x,y)=1\}$ is indicatrix of $F$ at $x$. I would like to know why induced Riemannian metric on indicatrix $I_F$ is in ...
2
votes
1answer
43 views

Need help understanding a relation between the fundamental forms

The book I am reading briefly mentions this relation between the fundamental forms but gives no explanation of how they got it. Take the following as the Weingarten Map/Shape Operator where $\nu$ is ...
1
vote
0answers
36 views

A problem possibly using the technique which has been used to prove the Whitney Embedding Theorem.

Problem. (1) Suppose $F:S^1 \to \mathbb{R}^3$ is a smooth immersion (i.e. $dF$ is nowhere zero). Prove that there is a unit vector $\mathbf v$ in $\mathbb{R}^3$ such that $\pi_{\mathbf v} \circ F : ...
0
votes
0answers
8 views

Proof of the categorisation of 1 dimensional connected differential manifolds, using the topological classification?

If we know that every connected, second countable topological 1-manifold is homeomorphic to the circle or the real line, is there a simple way to use it to prove the analogous statement for ...
0
votes
1answer
33 views

Length of closed curve

How to find length of this closed curve? I dont know what limits should i take for the integral.
0
votes
0answers
12 views

Geodesic equation applied to halfplane model

I have learned some things regarding connections and geodesic. And I want to apply this knowledge to the exercise: show that the vertical lines in the halfplane model are geodesics. The metric is ...
8
votes
1answer
38 views

Is it possible to have a sphere $S^m$ equidistant to sphere $S^n$ in $R^k$?

Is it possible to place a sphere $S^m$ and another sphere $S^n$ in Euclidean $k$-dimensional space $R^k$ in such a way that the distance from any point of the first sphere to any point of the second ...