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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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 ...
4
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18 views

Angle form, 1-form, proof verification.

Check that the $1$-form $d\,\text{arg}$ in $\mathbb{R}^2 - \{0\}$ is just the form$${{-y}\over{x^2 + y^2}}\,dx + {{x}\over{x^2 + y^2}}\,dy.$$ The following is my solution. Observe that we can ...
5
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1answer
55 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 ...
2
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24 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
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1answer
42 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
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1answer
28 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
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1answer
21 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 ...
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13 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, ...
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0answers
19 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
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5 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
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19 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 ...
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23 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 ...
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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
3answers
64 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 ...
4
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1answer
42 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
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1answer
22 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, ...
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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
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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
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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 ...
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1answer
22 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 ...
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1answer
25 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 ...
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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 ...
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1answer
45 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 $z=x+i y$.
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29 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
70 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
40 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
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1answer
32 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
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80 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 ...
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18 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 ...
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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
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0answers
24 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
42 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
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0answers
35 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
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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
11 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 ...
7
votes
1answer
37 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 ...
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+50

Prove that the sphere is the only closed surface in $\mathbb{R}^3$ that minimizes the surface area to volume ratio.

It is well known that a sphere minimizes the surface area to volume ratio since it reaches equality in the Isoperimetric Inequality. I'm trying to prove that no other closed surface has this property. ...
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0answers
41 views

Geometry of Curves

I found this question in question paper of Geometry of Curves and surfaces from Leeds University. Can anyone help me how I solve it.
1
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0answers
18 views

Gradient of second fundamental form

In the book I'm reading ("Differential Geometry Curves-Surfaces-Manifolds by Wolfgang Kuhnel") two definitions of prinicpal curvatures directions are presented: The extramum values of $II(X,X)$ ...
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0answers
24 views

Circumference of hyperbolic circle is $2\pi \sinh r$

I'm looking for a proof that in the Poincare disk model the circumference of a circle of radius $r$ is $2\pi \sinh r$. I have seen this result in many places but I haven't been able to find a proof. ...
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1answer
28 views

Confused with The Transversality Theorem when all manifolds are boundaryless

In Guillemin-Pollack's book Differential Topology, the Transversality theorem states that The transversility Theorem. Suppose that $F:X \times S \to Y$ is a smooth map of manifolds, where only $X$ ...
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21 views

Isometry of covering space [on hold]

Let $M$ be a compact Riemannian manifold. Consider a covering space $N$ of $M$, with the pull-back metric from $\pi : N \to M$. Given a point $x \in M$, and a couple of points $y, z \in \pi^{-1}(x) ...
2
votes
1answer
23 views

Expressing a differentiable map and curve in a parametrization

This is a question mainly about notation that I just cannot seem to understand. I'm reading Do Carmo's book "Riemannian Geometry" on page 7. Here is some context: (Here, $\alpha : (-\varepsilon, ...
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0answers
22 views

Exactly one supporting line for a $C^1$ Jordan curve [on hold]

Let $\gamma :[a,b]\to\mathbb{R}^2$ be a convex Jordan curve (closed, simple, continuous) that has $C^1$ regularity, with $\gamma '(t)\neq 0,\ \forall t\in [a,b]$. Prove that there is exactly one ...
2
votes
0answers
52 views

What's the most general geometry branch?

What is the most general geometry of curves and surfaces? For example, at curves, we define in differential geometry the tangent vector as the derivative of a regular curve, but visually many other ...
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1answer
45 views

Show $H_{dR}^1(S^n)=0$ for $n>1$ without de-Rham Theorem, and some similar questions.

Question Without using de Rham's theorem, prove: (1) Show $H_{dR}^1(S^n)=0$ for $n>1$. (2) Use (1) to show $H_{dR}^1(RP^n)=0$ (3) a n-form $\Omega$ is exact on $S^n$ if and only if $$ ...
2
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0answers
31 views

Try to use Homogeous space Characterize the space of all lines in the plane .

The problem is just to gives a smooth manifold structure of all straight lines in $\mathbb{R^2}$ ( not just those which pass through the oringin).Moreover, identify it with a well-known manifold! My ...