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)

3
votes
1answer
473 views

Showing the parametrically representation of hyperbolic paraboloid. And how to find the curves $u$ and $v$ be constant.

Show that the hyperbolic paraboloid can be represented parametrically as $$r(u,v)=\langle a(u+v), b(u-v), uv\rangle$$ Find the curves $u$ is constant and $v$ is constant. I guess I need to use the ...
1
vote
1answer
26 views

To show that $\Lambda^pL(V\rightarrow W)$ and $L(\Lambda^pV\rightarrow W)$ are not necessarily isomorphic

Let $V$ and $W$ be two vector spaces. Use $L(V\rightarrow W)$ to represent the vector space of linear map from $V$ to $W$. It is proved that $\Lambda^p(V^*)\cong (\Lambda^pV)^*$, where $\Lambda$ is ...
4
votes
1answer
109 views

The curve has constant torsion.

Question: Show that when the curve $c_1=c_1(t)$ has constant torsion $\tau$, the curve $$c_2=c_2(t)=-\frac{1}{\tau}N+\int_{t_0}^{t}B(u)du$$ has constant curvature $-\tau$ or $+\tau$. What I ...
3
votes
1answer
146 views

Covariant derivative with contravariant components derivation

I'm doing Leonard Susskind's course on General Relativity (http://deimos3.apple.com/WebObjects/Core.woa/Feed/itunes.stanford.edu-dz.19344853322.019344853324 ), and I'm stuck on a particular derivation ...
0
votes
1answer
157 views

Equivalent definitions of the tangent space

L.S., In my book Vector Analysis by Klaus Jänich, Three different 'versions' of the Tangent space of a point $p$ at a differentiable variety are being discussed. The 'geometrical': the set of ...
2
votes
0answers
52 views

Finding an isometry given the distance between points

If $A,B,C,D \in \mathbb{R}^n$, such that $d(A,B)=d(C,D)$, then exists an isometry $f:\mathbb{R}^n\rightarrow \mathbb{R}^n $, such that $f(A)=C, f(B)=D$. Thanks a lot for the help.
0
votes
1answer
149 views

$(\frac{1}{\kappa})^2+(\frac{\dot{\kappa}}{\kappa^2\tau})^2=r^2$

Show that for a curve lying on a sphere of radius r with nowhere vanishing torsion, the following equation is satisfied: $$(\frac{1}{\kappa})^2+(\frac{\dot{\kappa}}{\kappa^2\tau})^2=r^2$$ Please ...
2
votes
3answers
60 views

What separates rotations from other co-ordinate transformations?

I am confused about some seemingly elementary ideas. From what I have understood, a rotation is just a specific class of co-ordinate transformations. If this is true, what exactly separates a rotation ...
1
vote
1answer
218 views

Comprehensive references on partial differential equations

How do the three volumes by Taylor's "Partial differential equations" compare with the two volumes with the same title by Friedrich Sauvigny's as a reference for study? What are the good and bad ...
5
votes
2answers
501 views

The tangent space of a manifold at a point given as the kernel of the jacobian of a submersion

Let $\phi:M\to N$ is a smooth map, $q\in N$ a regular value, and $V=\phi^{-1}(q)$. I want to show that, for each $p\in V$, $T_p(V)= \mathrm{ker}(\phi_*)\subseteq T_p(M)$ (where $\phi_*$ is the ...
4
votes
1answer
61 views

Show that $\dot{n_s}=-\kappa_s t$

I found the question in a differential geometry textbook while studying. This question seems so intesting to me. So please help me solving it. Show that, if $\gamma$ is a unit-speed plane curve, ...
3
votes
1answer
81 views

Conformal mappings of non-orientable surfaces

Is it true that for any non-orientable Riemannian surface of genus 2 there exists Conformal mapping of degree two to a projective plane? Also, is the following argument works? Given any Riemann ...
3
votes
1answer
255 views

For a closed plane curve, showing some inequalities.

I have a problem following : Let $\gamma:[0,T]→\mathbb{R}^2$ be a closed plane curve, i.e., a regular parametrized curve such that $ \gamma$ and all its derivatives agree at 0 and $T$. For ...
3
votes
1answer
141 views

If $\gamma$ is spherical, then the equation $\frac{\tau}{\kappa}=\frac{d}{ds}(\frac{\dot{\kappa}}{\tau \kappa^2})$ holds.

Question: Let $\gamma (t)$ be a unit-speed curve with $\kappa(t)\gt0$ and $\tau(t)\neq0$ for all $t$. Show that, if $\gamma$ is spherical, i.e., if it lies on the surface of a sphere, then ...
6
votes
1answer
184 views

First Pontryagin class on real Grassmannian manifold?

I wonder if real Grassmannian manifold $SO(p+q)/SO(p) \times SO(q)$ have nontrivial first Pontryagin class? I only have physics background and know really little about characteristic class theory.
13
votes
1answer
987 views

Research in differential geometry

I am an 3rd year undergrad interested in mathematics and theoretical physics. I have been reading some classical differential geometry books and I want to pursue this subject further. I have three ...
8
votes
2answers
133 views

How to obtain $y$

The question was written with dark-blue pen. And I tried to solve this question. I obtained $x$ as it is below. But I cannot obtain $y$ Please show me how to do this. By the way, $\gamma (t)$ ...
2
votes
1answer
98 views

Verify that an ellipse has four vertices.

Verify that an ellipse has four vertices. The ellipse is given by $$ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ And I took $$x=a\cos t$$ and $$y=b \sin t$$ for $t\in [0,2\pi]$ Please can someone help ...
7
votes
4answers
497 views

Line bundles of the circle

Up to isomorphism, I think there exist only two line bundles of the circle: the trivial bundle (diffeomorphic to a cylinder) and a bundle that looks like to a Möbius band. Although it seems obvious ...
3
votes
1answer
95 views

How to compute this angle form integral?

Let $\gamma$ be the curve in $\Bbb{R}^2$ given by $x^2/9+y^2/4=1$ with counter-clockwise orientation. Compute $$\int_{\gamma} \frac{-ydx+xdy}{x^2+y^2}$$ I guess that the answer should be $2 \pi$ ...
2
votes
1answer
70 views

Fundamental group of a component of $GL_n({\bf R})$

Let $G$ be a component of $GL_n({\bf R})$ such that element has a positive determenant. (1) Since it contains $SO(n)$, $\pi_1(SO(n))$ ? What is a fundamental group of $G$ ? (2) It has a curvature ...
2
votes
1answer
64 views

Why isn't the partial derivative of a coordinate patch a vector field?

Let $D$ be an open set of $R^2$ and M a surface in $R^3$. Let $\mathbf{x(u,v)}: D \rightarrow M $ be a coordinate patch in M. Let $\mathbf{x}_u, \mathbf{x}_v$ be partial derivatives of the patch. ...
6
votes
0answers
384 views

Surface integral for a scalar function defined on a discrete surface

Imagine a polyhedral, discrete surface embedded in $\mathbb{R}^3$. Its faces are all triangles. For each vertex, one can compute the discrete mean and Gaussian curvatures and evaluate the sum of ...
3
votes
1answer
84 views

The non-vanishing 1-form on $\mathbb R^2$

If $\omega$ is a non-vanishing 1-form on $\mathbb R^2$, then for any a point $p\in \mathbb R^2$, can we find an open neighborhood $U$ of $p$ and two functions $f,g$ on $U$ such that $\omega=fdg$ on ...
7
votes
3answers
348 views

Why must this function have a critical point inside the sphere?

Suppose we have $f: \mathbf{R}^{3} \to \mathbf{R}$ with the following property: $\langle \nabla f(x), x \rangle > 0$ for every $x \in S^{2}$, that is, it's gradient points outwards the unit sphere. ...
2
votes
1answer
86 views

Laplacian on ${\bf R}^2$ and mean curvature

Consider a function $f$ on ${\bf R}^2$ whose critical point is origin. Then Gaussian curvature of graph of $f$ at origin is determinant of ${\rm Hess} \ f$ and Mean curvature is trace of ${\rm Hess} ...
0
votes
1answer
79 views

How to show a flow is a rotation of $R^{3}$

On $R^3$ let X,Y,Z be the vector fields $X=z\frac{\partial}{\partial y}-y\frac{\partial}{\partial z}$ $Y=-z\frac{\partial}{\partial x}+x\frac{\partial}{\partial z}$ $Z=y\frac{\partial}{\partial ...
7
votes
2answers
180 views

How do manifolds have enough structure to do calculus?

I am referring, of course, to to differentiable manifolds. I've seen a few different definitions. The one I like best is the one which says it's a topological space such that every point has a ...
5
votes
2answers
82 views

curvature of curve in $\mathbb R^2$

I'm taking a course in differential geometry this semester and I'm stuck with one of my first exercises. Let $\alpha(s)=(x(s),y(s))$ be a curve such that $|\alpha'(s)|=1$. Prove that the curvature is ...
0
votes
0answers
84 views

Observation on normalized Ricci flow on two sphere

Note that two sphere with nonnegative curvature converges to canonical sphere along normalized Ricci flow. So assume that if curvature is bounded below by $-\epsilon$ then the sphere meets a ...
3
votes
1answer
83 views

complete metric on a Riemann Surface

I'm reading a book and found a sentence I don't understand: Every Riemann surface $S$ supports an essentially unique complete metric of constant curvature $1$, $0$ or $-1$. Every point of ...
-1
votes
1answer
59 views

how to find the signed normal

$$\gamma (t)= (R\cos (t/R), R\sin (t/R))$$ $$\dot {\gamma (t)}=(-\sin (t/R), \cos (t/R))$$ $$n_s= (-\cos (t/R), -\sin (t/R))$$ where $n_s$ is the signed normal. the instructor has found the $n_s$. ...
5
votes
2answers
106 views

Why is connection a map from $\Gamma(E)$ to $\Omega^1\otimes\Gamma(E)$?

On the site Vector Bundle Connection, it gives two definitions of a connection. One is view a connection as a linear map from a section of $E\otimes TM$ to a section of $E$: $$ D:\Gamma(E\otimes ...
3
votes
0answers
115 views

Invariance of Integration on Homotopic Curves

All: I'm trying to show that if curves $\gamma, \gamma'$ are homotopic to each other in some region $R$ (open, connected subset) of the plane, and f is differentiable in $R$ , then: ...
2
votes
2answers
137 views

Gaussian curvaure of a surface revolution

Let $\alpha(s)=(f(s), g(s))$ be a plane curve parametrized by arc length on the $yz$-plane and assume that $f(s)>0.$ The surface revolution attained by rotating the curve parameterized by $\alpha$ ...
2
votes
1answer
119 views

derivative on manifold

Let $f: \mathbb{R}^n \to \mathbb{R}^n$ be a smooth function and $M$ be a smooth manifold of R^n. Assume that $Df(x)v \neq 0$ for all $v$ being tangent to $M$ at $x$ and for all $x$ in $M$. Can we say ...
5
votes
1answer
286 views

Holonomy and Differential Characters

This question is going to be rather vague, but I'm just trying to see if there are obvious connections between these two concepts. So the holonomy of a vector bundle with Lie group $G$ is ...
2
votes
1answer
97 views

Stokes' Theorem for the upper half space.

How can I prove Stokes' Theorem $\int_M dω = \int_{∂M} ω$ where $M = \mathbb{H}^ n$, the upper half space.
4
votes
0answers
144 views

Submanifold of a regular value of a manifold with boundary

Question: Suppose $M$ is a smooth manifold with boundary, $N$ is a smooth manifold, and $F:M\rightarrow N$ is a smooth map. Let $S=F^{-1}(c)$, where $c\in N$ is a regular value of both $F$ and ...
2
votes
0answers
70 views

Metric on the sphere involving tensor product

The metric on the sphere in the $(\theta,\phi)$ is of the form: $$ g_{\theta\theta}=r^2,g_{\theta\phi}=g_{\phi\theta}=0,g_{\phi\phi}=r^2\sin^2\theta $$ When transforming it to the $(x,y)$ coordinate ...
0
votes
1answer
105 views

Finding matrix for parallel transport map.

Consider the surface S given by the patch σ(u, v) = (u, v, 0) and the points p(0, 0, 0), q(1, 1, 0) ∈ S. Chose bases for TpS and TqS and write down the matrix for the parallel transport map Πγp,q ...
3
votes
1answer
209 views

A linear connection induces a covariant derivative of tensor fields.

Let $M$ be a smooth manifold. notation: $\mathcal T(M)^{(k,l)}$ is the $C^{\infty}(M)$-module of all tensor fields of type $(k,l)$ on $M$ ($k$ indicates the covariant part). $\mathcal ...
2
votes
2answers
110 views

How to calculate Frenet-Serret equations

How to calculate Frenet-Serret equations of the helix $$\gamma : \Bbb R \to \ \Bbb R^3$$ $$\gamma (s) =\left(\cos \left(\frac{s}{\sqrt 2}\right), \sin \left(\frac{s}{\sqrt 2}\right), ...
0
votes
1answer
3k views

how to calculate the curvature of an ellipse

how can I compute the curvative of an ellipse given by $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ do i need to take $x=acos(t)$ and $y=bsin(t)$? please show me a way how to solve this? thank you for ...
0
votes
0answers
39 views

An abstract definition of the cotangent space to a smooth manifold.

I need a book introducing the cotangent space to smooth manifolds in the most abstract way. So $T^\ast_p M$ by this point of view should be the quotient ring $I/I^2$ where $I=\{[f]\in ...
0
votes
2answers
56 views

the differential of a function $f\in C^{\infty}(M)$: two definitions

Let $M$ be a smooth $n$-dimensional manifold (on $\mathbb R$). If $p\in M$, we have that $\Big\{\frac{\partial}{\partial x^1}\Big|_p,\ldots,\frac{\partial}{\partial x^n}\Big|_p\Big\}$ is a basis for ...
1
vote
1answer
44 views

nonorientability of the projective plane by vanishing 2 form

I raised a related question but hope to get some answer using the nonvanishing 2 form definition. Let P be the real projective plane obtained by identifying antipodal points on the unit sphere of ...
1
vote
2answers
470 views

showing that two surface is isometric

I tried to show that a parametrized surface $S$ in $\mathbb{R}^3$ given by $(u, v)$ ->$(u, v, u^2)$ is isometric to the flat plane. At first, I found their first fundamental form, but they are ...
0
votes
1answer
91 views

Normal curvature of a circle in a plane

I have the circle $\gamma(t) = (\cos t, \sin t, 0)$ in the plane $z=0$. Now I understand that normal curvature is related to the second fundamental form, and an expression for it is ...
1
vote
2answers
116 views

Show that the vector field $X(x, y, z)=(xy-z^2, yz-x^2, x^2+z^2+xz-1)$ is tangent to the set $x^2 + y^2 + z^2 = 1$

I know I need to find functions $F(t)$, $G(t)$, and $H(t)$ such that $F(0)=x$, $G(0)=y$, and $H(0)=z$ and $F'(0)=xy-z^2$, $G'(0)=yz-x^2$, and $H'(0)=x^2+z^2+xz-1$. It's also necessary that $(F(t))^2 ...