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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Parametrization of surfaces gauss

What is pde relating f and g if the Gauss curvature is +1 and -1 respectively: (x,y,z) = ( u cos(v), u sin(v), f(u,v) ). (x,y,z) = ( u cos(v), u sin(v), g(u,v) ). I had tried and got results with ...
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What is the anticommutator of the interior product and codifferential (adjoint of exterior derivative)?

What is $\eta=i_X \delta + \delta i_X$ acting on differential forms? Here $\delta$ is the usual Hodge adjoint of the exterior derivative and $i_X$ is contraction of a form with the vector field $X$. ...
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Stereographic projection of a sphere

What should have been a simple exercise in geometry has morphed into a multi-day affair with me figuratively tearing my hair out. I have no clue what's wrong. This image accompanies the problem: ...
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45 views

Cauchy integral formula: can it be proved like this?

Consider the Cauchy theorem: Let $D\subset \mathbb C$ be a domain such that $\partial D$ is smooth and $\overline{D}$ is compact. Let $f$ be holomorphic on $D$ and continuous on the closure. Then ...
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Paramatrizes curve with constant speed

Show that if $\alpha : I \rightarrow \Re^{n+1}$ is a parametrised curve with constant speed then $\alpha(t) \perp \frac{d}{dt} \alpha(t)$ for all $t\epsilon I$.
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First encounters with sheaves: could you tell me if my thoughts are correct?

Let $X=\mathbb R$ be the reals and $\mathbb Z$ the integers (with the discrete topology). Do I understand correctly the definition of sheaf? Here is how I understand it (illustrated by a ...
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Why are derivations useful for defining tangent vectors?

On page 54 in his book Introduction to Smooth Manifolds, John Lee says the following: A linear map $v: C^\infty (M) \rightarrow \Bbb{R}$ is called a derivation at p if it satisfies ...
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39 views

Map between Tangent Manifolds Well-Defined?

Let $f: \mathcal{M} \to \mathcal{N}$ be a $\mathscr{C}^{r+1}$ map. We define a map $\mathscr{T}f: \mathscr{T}\mathcal{M} \to \mathscr{T}\mathcal{N}$ as follows: A local representation of the map ...
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24 views

critical point for the curvature does not correspond to a local maximum/minimum.

Draw an example where a critical point for the curvature does not correspond to a local maximum/minimum Does the curve for infinity sign satisfy this? I am having trouble seeing why it's true, if of ...
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If the curvature of a curve in a plane is a constant, is the curve contained in a circle?

If the curvature of a curve in a plane is a constant, is the curve contained in a circle?(Suppose curvature is positive.) one of my homework problems needs to use this, but I am not sure whether this ...
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Recover a surface's equation from its curvature

Can the equation of a surface in Euclidean 3 space be recovered from the equation of its Gaussian curvature?
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49 views

Closed non-exact $2$-form on $T = S^1 \times S^1$

I would like to find a closed non-exact $2$-form on $T = S^1 \times S^1$. Here are my thoughts: Since $d\theta$ and $d\varphi$ are closed non-exact $1$-forms an obvious candidate is $d\theta \wedge ...
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29 views

Finding two inequivalent closed, non-exact $1$-forms on $T = S^1 \times S^1$: second check

This is a follow up on my previous question. I would like to test whether I understand the first part of the answer given to me there by rewriting it in my own words. Please could someone tell me ...
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1answer
51 views

What class would this be covered in?

Would the material in this section of a wikipedia article be covered in a standard course on Differential Geometry, or should I look elsewhere to learn those sorts of things? Specifically, topics like ...
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47 views

Gluing submanifolds along their common boundary

Let $M$ be a smooth $m$-manifold and $N_1$, $N_2$ smooth embedded $k$-submanifolds such that $N_1\cap N_2=\partial N_1=\partial N_2$, for each $x\in N_1\cap N_2$, $T_x N_1=T_x N_2$, and for $x\in ...
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35 views

the top chern class of the holomorphic tangent bundle is the euler class

Is the following true? Let X be a complex manifold of complex dimension d and let V denote its holomorphic tangent bundle (ie it's $T^{1,0} \subset T \otimes_R C$, where T is the tangent bundle of ...
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Exterior Algebra: Characterization [on hold]

Just a short question: Given a vector space. Why is the exterior algebra characterized as: The largest anticommutative integer-graded algebra with identity linearly embedding the vector ...
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A doubly ruled surface which is not a plane must be quadratic

I want to show that a doubly ruled surface which is not plane must be quadratic. Any help will be appreciated.
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61 views

Pullback of a differential form

My question is in regards to a proof in Lee's 'Introduction to Smooth Manifolds'. He proves a lemma about the pullback of a differential form on a manifold $N$, where $F:M\rightarrow N$ is a smooth ...
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51 views

Prove that $\frac{dN}{ds}=-\kappa T$

Prove that $\frac{dN}{ds}=-\kappa T$, where $N$ is the oriented normal, and $T$ is the unit tangent vector, and $s$ is arc-length parameter. Here's what I've got so far from my note and I don't ...
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3answers
42 views

Embeddings are precisely proper injective immersions.

We call a map $f: X \to Y$ between topological spaces proper if $f^{-1}(K)$ is compact for all compact $K \subset Y$. Where can I find a reference that embeddings are precisely proper injective ...
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Question concerning tensors

As some of you may have seen from my previous question(s), I am working through Spivak's Calc on Manifolds and this happens to be the first time I've been introduced to tensors formally, though I have ...
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68 views

Finding two inequivalent closed, non-exact $1$-forms on $T = S^1 \times S^1$

I've been studying the torus and the first cohomology group $H^1_{dR}(T)$ for a couple of weeks now. I finally had a breakthrough of understanding and would like to kindly request the community to ...
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When is a linear map of 1-forms a pullback?

Every diffeomorphism $\phi: M\to N$ between two-dimensional compact oriented Riemannian manifolds induces a linear map on one-forms $L:\Omega^1(M)\to\Omega^1(N)$ given by the pullback of $\phi^{-1}$. ...
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What sort of algebraic structure describes the “tensor algebra” of tensors of mixed variance in differential geometry?

differential geometer in training here. With regards to my background, I learned differential and Riemannian geometry from O'Neill and Lee's series. I'm working on my algebra background (which is ...
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Is saying each pt of a topo. manifold has nbhd homeomorphic to R^n the same thing as saying there is a local coordinate system at each point?

Is saying each point of a topological manifold has a neighborhood homeomorphic to $\Bbb{R}^{n}$ the same thing as saying there is a local coordinate system at each point? I'm not really sure what ...
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47 views

Find a parametrization of the intersection curve between two surfaces in $\mathbb{R^3}$ $x^2+y^2+z^2=1$ and $x^2+y^2=x$.

Find a parametrization of the intersection curve between two surfaces in $\mathbb{R}^3$ $$x^2+y^2+z^2=1$$ and $$x^2+y^2=x.$$ I know that $x^2+y^2+z^2=1$ is a sphere and that $x^2+y^2=x$ is a circular ...
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58 views

Gauß and mean curvature

I was wondering whether the Gauß, mean curvature and shape operator of a surface actually depend on the chosen parametrization? Under a reparametrization of $f: \Omega \subset \mathbb{R}^2 ...
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56 views
+50

Isometric and conformal map

We defined conformal and isometric maps for surfaces $f,g: \Omega \subset \mathbb{R}^2 \rightarrow S \subset \mathbb{R}^3$. Under a reparametrization of $f$ I understand a diffeomorphism $\Phi : M ...
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Zero Gauss curvature surfaces spanning a loop

EDIT 1: Spanning across a given arbitrary closed boundary/loop a surface can be defined with zero mean curvature H in 3-space as minimal surfaces. Likewise can a surface be defined with zero Gauss ...
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Hessian of a function on Riemannian manifolds

Let $(M,g,\nabla)$ be a Riemannian manifold with metric $g$ and Riemannian connection $\nabla$. The hessian of a function $f:M\to R$ is defined by: $$H^f(X,Y)=g(\nabla_X\ \ \operatorname{grad} ...
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vertical/horizontal asymptotes - general understanding

Do vertical asymptote only exists in fractions? My taught was yes. Can the curve cross a vertical asymptote? My taught was no. Can the curve cross a horizontal asymptote? my taught was yes. Thanks
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Embed curves in the plane

The strongest version of Whitney's embedding theorem says that every smooth real $n$-dimensional manifold $M^n$ (Hausdorff and second-countable) can be embedded in $\mathbb{R}^{2n}$. This should mean ...
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1answer
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Must a Developable Surface be Tangent Developable or a Generalised Cone/Cylinder?

I've commonly seen that tangent developable surfaces, Generalised cones and generalised cylinders are developable surfaces. (see http://en.wikipedia.org/wiki/Developable_surface) But are these the ...
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Multivariable Calculus or Differential Geometry (Analysis on Manifolds) after single variable calculus

Background: Applied Mathematics program, finished with single variable calculus, and in parallel with basic analysis. (Not knowledge of multivariable calculus yet) Please feel free to recommend ...
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How to evaluate this integral: $\oint dx$?

I am trying to understand differential forms. Now I tried to evaluate $$ \oint_{S^1}dx$$ I should get anything non-zero but I don't know how to do it (even though I know the result). If $S^1$ in ...
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Use Frenet Frame and Pythagorean Theorem.

Suppose we have a curve $c(t)$ where $t$ goes from $a$ to $b$. $c$ has positive curvature and a frenet frame(TNB). Choose $\rho > 0$ and small. and define: $f(t) = c(t) +\rho B(t)$ $g(t) = c(t) ...
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Basic Differential geometry: Shortest path between two points in R^3 is straight.

Given two points P and Q in $\mathbb{R^3}$, we want to show that the shortest distance between them is through a straight line. let $c(a) = P$ and $c(b) = Q$ and $c(t)\neq P$ for $t>a$(One ...
2
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1answer
46 views

Relationship between differential forms and coordinates

For the purpose of this question let us restrict our considerations to smooth $3$-manifolds. So the manifold $M$ we consider here is endowed with smooth coordinate charts $(x,y,z)$. What I have ...
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37 views

Embedded Submanifolds Have a Unique Smooth Structure

Let $M$ be a smooth manifold. An embedded submanifold of $M$ is a subset $S$ of $M$ such that $S$ is a topological manifold under the subspace topology induced by $M$, endowed with a smooth structure ...
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43 views

Is there a better way to show the intrinsic curvature of a cylinder is zero?

I am new to differential geometry and Riemannian geometry. I have on two separate occasions (separated by 6 months) encountered exercises where I feel like I am not giving a complete answer. ...
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1answer
28 views

Sum of Killing vector fields is a Killing vector field

Let $(M,g)$ be a Riemannian manifold. A smooth vector field $X$ is called a Killing vector field if the flow of $X$ acts by isometries, or, equivalently, if $L_X g = 0$. Now why is the sum of Killing ...
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Laplace's equation in rectangle geometry

Consider Laplace's equation in a rectangle with length and width of a and b respectively, with following boundary conditions: All the boundaries with $x < a/2$ have Drichlet boundary condition ...
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How many isometries of the (unit) 2-sphere are there?

I had a homework problem that exhibited two Killing vector fields for the 2-sphere, asked me to find a third and then asked me if there are any more. I answered no because the Lie Algebra of the ...
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Compute a parallel transport

Let $\mathbb{S}^{2} \subset \mathbb{R}^{3}$ be the $2$-sphere ($\mathbb{S}^{2} = \left\{ (x,y,z) \in \mathbb{R}^3, \; x^2+y^2+z^2 = 1 \right\}$). Let $p \in \mathbb{S}^{2}$ and $\xi \in T_{p}S^{2} = ...
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How to prove easily that geodesic is auto parallel?

I only have the elementary concept of geodesic and differential equn of geodesic. Intrinsic derivative =0 implies parallel displacement along a curve. What does it mean by auto parallel? How to prove ...
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Differentiation on Manifolds Basics

I'm having some real trouble comprehending integral curves and Lie derivatives on a Manifold. I will write out my understanding and ask the questions below. For a vector field $X$ on smooth manifold ...
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Are there two kinds of Christoffel symbols?

I am struggling to understand Christoffel symbols. Part of my confusion is that there are two kinds. So I mix up which properties belong to each and end up learning about neither. Can someone define ...
2
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2answers
111 views

The space $x^3-y^2=0$

Consider $\{(x,y)\in\mathbf{R}^2 \ | \ x^3-y^2=0\}$ as a subspace of $\mathbf{R}^2$. Intuitvely I understand that this is not supposed to be a differentiable manifold because it has a cusp at $0$. But ...
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Transversality of leaves to the spheres .

Consider a form in the complex plane such that its linear part is $\omega_0=\lambda_1xdy-\lambda_2ydx$ in the Poincare domain: $\lambda_1\lambda_2 \ne 0$ and $\lambda_1/\lambda_2 \notin \mathbb{R}^-$. ...