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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296 views

Frenet-Serret to show something lies in the plane [duplicate]

Possible Duplicate: Prove that curve with zero torsion is planar How can we use the Frenet-Serret formulas to prove if $\alpha(s)$ is a unit speed curve with $\kappa \neq 0$ and $\tau = 0$, ...
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0answers
214 views

If $F$ is diffeomorphism the linear map $F_*$ is an isomorphism?

Let $f\colon M \rightarrow N$ be a smooth map. If $f$ is a diffeomorphism I am trying to show that the linear map $f_*$ : $T_pM \rightarrow T_{f(p)}M$ is an isomorphism for all $p \in M$. I know the ...
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1answer
103 views

Hippopede Parametrization

Given the following sphere and cylinder, $$\begin{align} x^2+y^2+z^2&=4R^2,\\ (x-R)^2+y^2&=R^2, \end{align}$$ find a parametric equation of their intersection. I know that their ...
7
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3answers
447 views

Embedding compact (boundaryless?) n-manifolds in n-dimensional real space

I know the embedding theorems that allow you to embed $n$-manifolds into $\mathbb{R}^k$, provided $k$ is chosen large enough. Here I'm interested in the possibility of taking $k=n$ in the case of ...
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1answer
107 views

What is the defined direction of the curvature vector at a point?

I'm asking this only because I could not find it anywhere on the web (tried Wikipedia and Google searches). The only hint I found was in this image on Wikipedia, which seems to indicate that the ...
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1answer
86 views

geodesics and differential equation satisfied

If $\gamma(s) = \mathbf{x}(\gamma^1(s),\gamma^2(s))$ where $\mathbf{x}$ is a coordinate patch, then what is the differential equations that $\gamma^k$ ($k = 1,2$) must satisfy if $\gamma$ is a ...
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0answers
122 views

horizontal vector in tangent bundle

I have a question about Do Carmo notion of horizontal vector (page 79). So he defines natural metric on $TM$ of manifold $M$. Now he chooses vector $V\in T_{(p,v)}(TM)$ and calls $V$ horizontal vector ...
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65 views

fixed point on a manifold

Suppose we have a Riemannian manifold $M$ with an open subset $U$ and a smooth map $\theta: U \to M$. If there is point $q\in U$ such that $\theta(q)=q$ prove that $d\theta_q=Id$ as a map ...
3
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1answer
129 views

Deciding whether a given set is a regular surface

I'm trying to decide whether the following set is a regular surface: $$\{(x^3 - 3xy^2, 3x^2y - y^3, 0) : (x, y) \in \mathbb{R}^2\}$$ I know that the map ${\bf x} : \mathbb{R^2} \to \mathbb{R^3}$ ...
2
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2answers
307 views

Tangent space to a manifold - First Order Approximation to the Manifold

I've a doubt about the tangent space to a manifold. Let $M$ be a $n$-manifold and let $p\in M$, I've heard that the tangent space $T_pM$ at $p$ is the first order approximation of $M$ near $p$ in the ...
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199 views

Nowhere vanishing magnetic helicity

Suppose you are given a nowhere-vanishing exact 2-form $B=dA$ on an open, connected domain $D\subset\mathbb{R}^3$. I'd like to think of $B$ as a magnetic field. Consider the product $H(A)=A\wedge ...
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1answer
58 views

Practise with Smooth Functions and Manifolds

I'm trying to get an intuition for smooth manifolds, and in particular the smoothness of transition functions. I haven't done that much calculus on $\mathbb{R}^n$ before, and would like to practice ...
3
votes
2answers
319 views

Christoffel symbols and fundamental forms

How can we prove that the christoffel symbol is \[ \Gamma^k_{ij} = \frac 12 \sum_{l=1}^2 g^{kl} \left(\frac{\partial g_{il}}{\partial u^j} + \frac{\partial g_{jl}}{\partial u^i} - \frac{\partial ...
3
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1answer
347 views

Pushforward of Inverse Map around the identity?

Let $G$ be a Lie group and $i:G \rightarrow G$ denote the inversion map. (Notation: $f_*$ is the pushforward map $F_*:T_pG \rightarrow T_{i(p)}G$ which takes $(F_{*}X)(f)=X(f\circ F)$ and $X$ is a ...
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1answer
144 views

geodesics and unit speed curves

Say we have 2 surfaces $M$ and $\hat M$ that intersect perpendicularly --> $\left<n,\hat n\right> = 0$ along the curve of the intersection intersection, where $n$ is the unit normal to $M$ and ...
0
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1answer
165 views

open interval in definition of curve

This is the second soft question I am asking today, so I apologise for that. This question, though probably a bit silly has been bugging me for a while and I have not come up with a satisfactory ...
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5answers
4k views

Exterior Derivative vs. Covariant Derivative vs. Lie Derivative

In differential geometry, there are several notions of differentiation, namely: Exterior Derivative, $d$ Covariant Derivative/Connection, $\nabla$ Lie Derivative, $\mathcal{L}$. I have listed them ...
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1answer
675 views

signed curvature

If we let $\alpha(s) = (x(s), y(s))$ be a unit speed curve in the plane, how can we define, $(t, n, k)$ where $k$ is the signed curvature? ii) Also, I need to prove that $t' = kn$, and $n = -kt$. I ...
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2answers
179 views

unit speed curves and frenet serret

Let us assume that $\alpha(s)$ is a unit speed curve with $\kappa > 0$. I'm trying to find the vector function $w(s)$ such that $$T' = w \times T,\quad N' = w \times N,\quad B' = w \times B.$$ I ...
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1answer
291 views

geodesics in differential geometry

Let gamma be a straight line in a surface M. How can we prove that gamma is a geodesic? ALl I note is that a geodesic on a surface M is a unit speed curve on M with geodesic curvature = 0 everywhere. ...
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192 views

Quotient map of the complex projective space

It's not to hard to see that the quotient map $\pi\colon \mathrm{C}^{n+1}\backslash \{0\} \to \mathbb CP^n$ is smooth and surjective. Does that imply that it is a submersion as well?
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0answers
138 views

A question about a definition of Ricci curvature

Let me first quote the definition of Ricci curvature from Wikipedia. Let $M$ be an n-dimensional Riemannian manifold equipped with its Levi-Civita connection $\nabla$. The Riemannian curvature tensor ...
3
votes
1answer
47 views

Possible lengths of geodecics

Let $M$ be a compact manifold (w/o boundary). Suppose that there is no closed geodesics on $M$ of length precisely $C$. I am trying to prove that there is an open cover $\{U_j\}$ of $M$ and $\epsilon ...
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1answer
137 views

Some manipulations on a Riemannian manifold

Let $\phi$ be a point on a Riemannian manifold $(M,g)$ and $\xi \in T_\phi M$. Then I want to understand the proof/meaning of the following three identities I ran into, $\nabla _\nu \xi ^i = ...
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0answers
280 views

compact manifolds are complete?

I know that every compact riemannian manifold $M$ is complete. And by Hopf-Rinow $M$ is geodesically complete. But now I'm confused by the closed unit disc $B^2\subset\mathbb{R}^2$. (I think $B^2$ is ...
3
votes
1answer
176 views

Invertible Derivative

I'm trying to brush up on some differential geometry, but there's a subtle point I don't understand. Suppose $h$ is a diffeomorphism. Then the lecture notes here suggest that it's derivative $df_x$ is ...
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0answers
229 views

Christoffel symbols

I found this equation $$\sum_{k,l,r,s} g_{ik}\frac{\partial g_{jl}}{\partial x^{r}}g^{rk}g^{sl}\frac{\partial}{\partial x^{s}}=2\sum_{s}\Gamma_{ij}^{s}\frac{\partial}{\partial x^{s}}$$ here g's are ...
2
votes
0answers
107 views

vector field on tangent bundle

I am studying geodesic flows on manifolds and I have problem visualizing it. How should one think about geodesic vector field on tangent bundle? How can you visualize it? Any intuition is welcome. ...
4
votes
1answer
247 views

Existence of Complex Structures on Complex Vector Bundles

Let $E$ be a real vector bundle on a smooth manifold $X$. Let $J : E \to E$ be a vector bundle morphism (i.e. $\pi \circ J = \pi$, where $\pi : E \to X$ is the projection map) with $J^2 = ...
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1answer
83 views

Square Roots of Hyperbolic Functions

I have the following vector function: $$ r(t)=(t-\sinh t\cosh t)\,\partial_x+2\cosh t\,\partial_y. $$ I computed its velocity to be as such: $$ r'(t)=-2\sinh^2t\,\partial_x+2\sinh t\,\partial_y. $$ ...
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0answers
144 views

The index of zero of vector field is well defined

Let $X$ be a vector field on a manifold $M$ of dimension $m$ (compact, connected, oriented) with isolated zero $z$. Let $B$ be a ball around $z$ such that $X$ has no other zeroes in $B$. We define the ...
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1answer
109 views

Definition of differential of Adjoint representation of Lie Group

Let $g$ be an element of Lie Group $G$, and $\gamma(t) : \mathbb{R} \rightarrow G$ be a path in $G$ such that $\gamma(0) = e$, the identity element of $G$. Denote the tangent space at $e$ as $T_eG$, ...
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1answer
110 views

Birational geometry as local algebraic geometry

Technically birational geometry is local geometry of algebraic varieties, yet it feels completely different from local differential geometry, which is more or less trivial. Is there some subtle ...
7
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1answer
311 views

Visualizing Exterior Derivative

How do you visualize the exterior derivative of differential forms? I imagine differential forms to be some sort of (oriented) line segments, areas, volumes etc. That is if I imagine a two-form, I ...
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0answers
122 views

Complex projective space CP3 (Twistor space), as bundle space with base CP1, and fiber 4-D Minkowski space-time?

Twistor space, as complex projective space $CP3$, is related to Minkowski 4-D space time (metric $1, -1,-1,-1)$, by the incidence relation. Let $Z = (v_a, u^{\dot{a}})$ a point in $CP3$, where $v_a$ ...
2
votes
3answers
755 views

Parametrization for the ellipsoids

Can someone help to describe some possible parametrizations for the ellipsoid: $$\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1?$$ I am thinking polar coordinates, but there may be the ...
6
votes
1answer
133 views

Complex structure on cotangent bundle

If M is a complex manifold with complex structure J, why the cotangent bundle of M carries a natural complex structure, and not an almost complex structure. Is that obvious?
1
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1answer
423 views

parallel transport preserves orientation

In my text its written that parallel transport on a Riemannian manifold preserves orientation. Can someone clarify what does that mean? I am confused about this notion.
0
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1answer
113 views

Central Projection of the Sphere

Given the parameterization of the unit sphere $x^2+y^2+z^2=1$ as $x = \displaystyle\frac{u}{\sqrt{1+u^2+v^2}} $ $y = \displaystyle\frac{v}{\sqrt{1+u^2+v^2}} $ $z = ...
2
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0answers
180 views

Fundamental vector fields

I have a question related to fundamental vector fields. For that I first setup the notations and properties etc. Let $G$ be a lie group acting smoothly on the manifold $M$. Let $\mathfrak{g}$ be its ...
4
votes
2answers
408 views

Do diffeomorphisms act transitively on a manifold?

Let $M$ be a smooth manifold, $x,y\in M$. Must there exist a diffeomorphism $f : M \rightarrow M$ with $f(x) = y$? I tried proving this via vector fields, i.e. trying to find a vector field whose ...
2
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0answers
56 views

Reconstructing paths on the sphere from the ratio of acceleration to velocity

Given a path $\gamma:[0,1]\to \mathbb C$, we can determine $\gamma$ from information about its derivatives. For example, $\gamma$ is determined by $\gamma(0), \gamma'(0)$, and $\gamma''(t)$. This ...
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1answer
85 views

Find an ellipse whose length is the same as the outer rim of the monkey saddle

Given the monkey saddle $z=x^3-3xy^2$ over the unit circle $x^2+y^2 \leq 1$, find an ellipse whose length is the same as the length of the outer edge of the monkey saddle. I've already found a ...
2
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1answer
139 views

condition for compatible connection on a Riemannian manifold

Prove that connection $\nabla $ on a Riemannian manifold $M$ is compatible with metric iff $$Xg(Y,Z)=g(\nabla_XY,Z)+g(Y,\nabla_XZ),$$ for every smooth vector fields $X,Y,Z$. I am confused about ...
3
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1answer
133 views

real part of a holomorphic function from a PDE

I have some problem that I can't figure out myself. Hope that someone can help me out. The problem is: Let $f : U \to \mathbb{R}$ be some real function on a simply-connected domain $U \subset ...
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1answer
264 views

Surface of revolution in differential geometry

The coordinate patch of the surface $x(t, \theta) = (r(t)\cos\theta, r(t)\sin\theta, z(t))$ doesn't cover the entire surface of revolution, ommitting points that would correspond to $\theta = \pm\pi$. ...
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vote
1answer
207 views

Find arc length of a circle using a hyperbolic metric

Given the hyperbolic metric $ds^2=\frac{dx^2+dy^2}{x^2}$ on the half plane $x > 0$, find the length of the arc of the circle $x^2+y^2=1$ from $(\cos\alpha,\sin\alpha)$ to $(\cos \beta, ...
2
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1answer
374 views

Leibniz rule for exterior derivative of a contraction

If I have a contraction of a vector field with a 1-form valued 2-form, what would be the appropiate product rule? $$d_{\left[a\right.} \left(P_{[bc]i} v^i \right)_{\left. \right]} = \, ?$$ This ...
7
votes
1answer
266 views

Total mean curvature in $L^2$ and minimal surfaces in spaces with non-positive sectional curvature

Let's suppose we have a Riemannian $n$-manifold $(N,g)$ and an immersed surface $f:\Sigma\rightarrow N$, with genus zero, equipped with the induced metric. Let's further assume that the ambient space ...
3
votes
1answer
66 views

Is there any loss in generality when I assume that a regular curve is arc-lenght parameterized?

My doubt is simple as that. When I have a smooth, regular curve (that is, its curvature is never zero), can I just assume that it is parameterized by arc-lenght, without any loss of generality? If ...