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
1answer
23 views

Parabolic points and curvature.

I have problems to solve this exercise: Let $p$ be a point of an oriented surface $S$ and assume that there is a neighborhood $U$ of $p$ in $S$ all points of which are parabolic. Prove that the ...
2
votes
0answers
17 views

Frenet frame and tangent space.

Let $\gamma(s) \subset \Sigma \subset \mathbb{R}^3$ a parametrized curve by arc lenght. Let suppose that $\gamma$ is on an oriented surface $\Sigma \subset \mathbb{R}^3$. We can consider the Frenet ...
3
votes
1answer
58 views

The Poincaré dual of a space-time curve

We have a smooth space-time curve defined by $f:C{\mapsto}M$, where $M$ is a typical curved space-time manifold. ${\eta}^{(4)}$ is the volume 4-form defined on $M$ and ${\varepsilon}^{(1)}$ is the ...
2
votes
1answer
46 views

Diffeomorphism between a sphere and ellipsoid in $\mathbb R^3$.

Diffeomorphism between a sphere and ellipsoid in $\mathbb R^3$. I've been looking for an diffeomorphism between a sphere in $\mathbb R^3$ and an ellipsoid of the form $$\{ (x,y,z) \in \mathbb R^3 ...
0
votes
0answers
29 views

Deformation theory in a paper of Bogomolov and Tschinkel

I am trying to read this paper http://www.math.nyu.edu/~tschinke/papers/yuri/00ajm/ajm.pdf, by Bogomolov and Tschinkel. I had 0 preparation in the theory of deformation of complex structures, but in ...
1
vote
1answer
33 views

Chronology condition and metric perturbations

Let $(M,g)$ be the quotient of the 2-dimensional Minkowski space-time by the discrete group of isometries generated by the map $f(t, x) = (t + 1, x + 1)$. Show that $(M, g)$ satisfies the ...
0
votes
1answer
29 views

Extremal length of a compound object

Lets say we want to find the extremal length of the family of curves $\Gamma$, which we say is $\mathcal{L}(\Gamma)$. $\Gamma$ moves through two adjacent Riemann manifolds, which have different ...
1
vote
1answer
39 views

Let $S$ be a regular surface and let $F$ be a diffeomorphism. How can I prove that the image $F(S)$ is a regular surface?

Let $S$ be a regular surface and let $F$ be a diffeomorphism. How can I prove that the image $F(S)$ is a regular surface ? Consider a regular surface $S \subset \mathbb R^3$ and a diffeomorphism ...
1
vote
0answers
49 views

Proof of some inquality on convex sets in $\mathbb{R}^n$

Suppose $\Omega$ is a strictly convex bounded set with smooth boundary in $\mathbb{R}^n$. If $y \in \partial\Omega$, how can we show that $$\langle(x-y),\nu_y\rangle < 0$$ for all $x \in ...
2
votes
0answers
43 views

Metric tensors and linear (differential) operators defined on a manifold and its osculating sphere

Given a 1D Riemannian manifold $\Gamma$ embedded in 2D Euclidean space (e.g. a parametric curve on a plane $\mathbb{R}^{2}$ ), and point $x_{0}\in \Gamma$, we denote $S^{1}(x_{0})$ the circle ...
1
vote
1answer
24 views

Gauge transformation on a principal bundle

I am reading through lecture notes found here and on pg 11 they define a map $\overline{\phi}_{\alpha}:\pi^{-1}(U_{\alpha})\rightarrow G$ by ...
1
vote
0answers
28 views

When the associated bundle to a $U(1)$-bundle is an holomorphic line bundle?

Let $P$ be a principal $U(1)$ bundle over a complex manifold $M$, and let $\rho\colon U(1)\to Aut(\mathbb{C})$ be the representation of $U(1)$ on $\mathbb{C}$ given by the standard multiplication. My ...
1
vote
1answer
34 views

How many degrees of freedom are in a flat metric and how does one count them?

I think that there are zero degrees of freedom in a flat metric, but I do not know how to count them. I know that any symmetric metric tensor has $n(n-1)/2$ degrees of freedom, where $n$ is the ...
3
votes
1answer
59 views

Ricci curvature of the Grassmannian?

Let $G(k, \mathbb{C}^n)$ be the Grassmannian of $k-$dimensional complex linear subspaces of $\mathbb{C}^n.$ We know that the Grassmannian can be embedded to the projective space ...
1
vote
3answers
28 views

How do I prove that there are only two possible orthonormal basis?

Let $x,y\in\mathbb{R}^3$ be unit vectors such that $\{x,y,(0,0,1)\}$ is an orthonormal subset of $\mathbb{R}^3$. Let $z\in \mathbb{R}^3$ such that $\{x,y,z\}$ is an orthonormal subset of ...
0
votes
1answer
22 views

Partion the boundary of a $n$-dimensional ball and write each partition as the graph of a $C^1$-function on a open subset of $\mathbb{R}^{n-1}$

Let $$S:=\left\{x\in\mathbb{R}^n:\left\|x\right\|_2\le r\right\}$$ How can we partition the boundary $\partial S$ of $S$ and write each partition as the graph of a continuously differentiable function ...
-2
votes
0answers
28 views

3d animation of a marble inside ellipsoid

I am new to MAPLE, I have been using Mathematica mostly. Here is what i am trying to do in MAPLE, Use the procedure plotmotion 2 on the plotmo worksheet to animate the motion of a marble in a bowl ...
1
vote
1answer
18 views

Is $|K/\tau|\leq 1$?

Let $\alpha:[a,b]\rightarrow \mathbb{R}^3$ be a regular curve such that $\alpha'$ and $\alpha''$ are linearly independent over $[a,b]$. Let $K$ and $\tau$ be the curvature and torsion of $\alpha$ ...
2
votes
1answer
43 views

Constant torsion-expression of unit speed curves

I am currently studying for an exam in differential geometry. There's a problem which I am not able to solve and do not even know where to start (although I think it has to do with the Frenet ...
8
votes
2answers
101 views

Learning modern differential geometry before curves and surfaces

What might one miss by learning modern differential geometry without first learning about curves and surfaces? I'm currently reading this book on differential geometry which starts with manifolds and ...
0
votes
1answer
26 views

Bound on the surface integral of the absolute Gaussian curvature given by surface of unit sphere?

im currently looking at an application in which I have to calculate or at least approximate the surface integral of the absolute Gaussian curvature over a patch of a regular surface $S$ given by a ...
1
vote
2answers
208 views

Some topics in Differential Geometry for a beginner

So our instructor wants us to write a 3 page report on some fact in Differential Geometry relating to curves or surfaces. It can be a theorem, a fact or some special case. The only restriction is ...
70
votes
9answers
19k views

Teaching myself differential topology and differential geometry

I have a hazy notion of some stuff in differential geometry and a better, but still not quite rigorous understanding of basics of differential topology. I have decided to fix this lacuna once for ...
10
votes
4answers
1k views

recommending books for intro to diff. geometry

I was wondering if anyone could recommend some books for studying topics such as abstract manifolds, differential forms on manifolds, integration of differential forms, stoke's thm, dRham chomology, ...
4
votes
1answer
52 views

Can some one help me parametrize $\frac{x^4}{a^4}+\frac{y^4}{b^4}+\frac{z^4}{c^4}=1$

Given a surface $$\frac{x^4}{a^4}+\frac{y^4}{b^4}+\frac{z^4}{c^4}=1$$how can I parametrize the surface using $X(u,v).$ I tried to use $$x=a\sqrt{\cos(\theta)\sin(\phi)}$$ ...
2
votes
1answer
51 views

Show that the meromorphic differential of the homogeneous polynomial is holomorphic and not isomorphic to $\mathbb{P_1}$

Consider the elliptic curve i.e. non-singular cubic, $X$ given by the equation $\xi_0\xi_2^2=\xi_1^3-\xi_0^2\xi_1$ in projective coordinates $(\xi_0:\xi_1:\xi_2)$, or, equivalently, by the equation ...
1
vote
0answers
46 views

How can I prove $dz=dx+idy$?

Let's see $\Bbb C$ as an $\Bbb R$-vector space. Hence it is isomorphic to $\Bbb R^2$ and it has dimension $2$. If $v_1,v_2$ is a basis for $\Bbb R^2$, every its element can be written as $xv_1+yv_2$; ...
3
votes
0answers
41 views

Differential Form Pullback Definition

I'm having some difficulty following how Spivak (Calculus on Manifolds) has induced the pullback on page 89. From reading elsewhere online it seems convention is to define the induced map of the ...
2
votes
1answer
65 views

Divisor of the meromorphic differential $\omega=\frac{dx}{y^3}$ on C: $\xi_1^4+\xi_2^4=\xi_0^4$

Consider Fermat's curve of degree 4 defined by C : $\xi_1^4+\xi_2^4=\xi_0^4$ in projective coordinates $(\xi_0 :\xi_1 :\xi_2)$ or, equivalently, by the affine equation $x^4 + y^4 = 1$ in the affine ...
0
votes
0answers
22 views

what does it mean that “not proportional”?

I'm reading this article. And here it says, "Let r(t) be a curve in Euclidean space, representing the position vector of the particle as a function of time. The Frenet–Serret formulas apply to curves ...
1
vote
1answer
37 views

Mean curvature of a ruled surface in $\mathbb{R}^3$

I was trying to prove that non compact surfaces with constant mean and gaussian curvature in $\mathbb R^3$ are part of planes or cylinder. Here is how I worked out the problem so far, First of all I ...
1
vote
1answer
24 views

Family of surfaces given their curvature

Let's suppose we know the shape operator (aka Weingarten operator) of a given surface everywhere in its domain. Is there any way, analytical or numerical, to find the family of surfaces having the ...
2
votes
1answer
42 views

Different definitions of tangent vector

I'm taking general relativity at the moment, and today in class the instructor gave us a definition of tangent vector as: $v$ is a tangent vector based at $p\in M$ if $v_{p}$ is a linear ...
1
vote
0answers
28 views

Derivative of the Gauss map is zero

If the derivative of the Gauss map is zero in every point in the image of a given local chart, can I conclude that the normal vector is constant and such image is contained in a plane? Edit: The ...
2
votes
1answer
39 views

Relationsip between two definitions of the christoffel symbol?

When I first started learing about tensor calculus, the professor defined the Christoffel symbol as $$\Gamma ^a _{bc} = Z^a \cdot \partial_b Z_c $$ Where $Z^a$ is a contravariant basis vector and ...
4
votes
2answers
78 views

How do I understand constraints on high order derivatives of the Gauss Map?

I'm trying to understand the constraints resulting from differentiating an unit normal field $N$ on a surface $S$ in $\mathbb{R}^3$. If I write the unit-length constraint at a point $p \in S$, I ...
1
vote
1answer
124 views

Find all unit speed planar curves $\alpha(s)$ such that the angle between $\alpha$ and $\alpha'$ is constant

I was trying to study for an exam in differential geometry and got stuck on the following problem : Determine all the planar curves $\alpha(s)$ parametrized by arc length, such that the angle between ...
3
votes
2answers
62 views

$\Delta_L(\text{im}\,\delta^*_g)\subset\text{im}\,\delta^*_g$ and $\Delta_L\big(\text{ker}\,\text{Bian}(g)\big)\subset\text{ker}\,\text{Bian}(g)$?

Let $(M,g)$ be an Einstein manifold with Levi-Civita connection $\nabla$ and whose Ricci tensor $\text{Rc}(g)=g$, in components $R_{ij}=g_{ij}$. The Lichnerowicz Laplacian of $g$ is the map ...
2
votes
1answer
39 views

Conifolds and Exotic Spheres

First of all, a disclaimer: I'm a physicist trying to understand mathematical aspects of some solutions I've encountered while studying string theory, and am certainly not a mathematician of any sort. ...
2
votes
2answers
867 views

Frenet-Serret formula proof

Prove that $$\textbf{r}''' = [s'''-\kappa^2(s')^3]\textbf{ T } + [3\kappa s's''+\kappa'(s')^2]\textbf{ N }+\kappa \hspace{1mm}\tau (s')^3\textbf{B}.$$ What is $\tau$, I can't figure that part out. ...
0
votes
2answers
113 views

Integration over manifolds

S is subset of $\Bbb R^3 $ consisting of the union of 1) $z$ axis 2) the unit circle $x^2+y^2=1,z=0$ 3) the points $(0,y,0)$ with $y \ge1 $ Let $A$ be the open set $\Bbb R^3-S$. Let $C_1, C_2, ...
5
votes
1answer
47 views

Intuition behind the connection one-form

When we define a connection on one principal bundle $\pi: P\to M$ with structure group $G$ we define it as an association of one subspace $H_pP\subset T_pP$ for each $p\in P$ such that $T_pP = H_pP ...
5
votes
1answer
77 views

What is the geometric interpretation of the Koszul formula?

I saw this simple form of Koszul formula on a book: $$2\ g(\nabla_XY,Z) = \mathcal{L}_Yg(X,Z) + (d\theta_Y)(X,Z)$$ where $\theta_Y$ is the one-form $g(Y,\cdot)$. It is equivalent to the more ...
1
vote
1answer
18 views

Non-vanishing normal vector field on a closed 3D curve

Let $C$ be a smooth regular simple closed curve in $\mathbb{R}^3$. Does there always exist a smooth non-vanishing normal vector field on $C$? I.e. a smooth $n: C \to \mathbb{R}^3 \setminus \{0\}$ such ...
1
vote
1answer
53 views

Is the tensor product of a complex line bundle with itself trivial?

Let $\xi$ be a complex line bundle over a manifold $M$. Then $\xi\otimes \xi$ is a trivial complex line bundle. Is my statement right?
2
votes
1answer
95 views

Very confused about directional derivatives as vectors

I'm currently reading about GR and how partial derivatives are basis vectors for a manifold, but I'm confused about how $dx^μ$ is the dual basis to $(\partial /\partial x)^μ$. I can't see how ...
1
vote
0answers
97 views

Vector fields as section of tangent bundle

We can define vector fields on manifolds in two ways. The way I first saw was that a vector field was a linear map $C^\infty(M) \to C^\infty(M)$ satisfying the Leibniz rule (aka product rule). We can ...
1
vote
0answers
28 views

Invertible Matrices as a Manifold [duplicate]

I've been trying to prove that the set of all invertible $n \times n$ matrices is a differentiable manifold. My attempt is as follows: Define a map $\alpha : X \to XX^{-1} - I$ I take the inverse: ...
-1
votes
0answers
44 views

Heat semigroup estimate on complete Riemannian manifold

Consider a complete noncompact Riemannian manifold $M$ such that the heat kernel $h_t(x, y)$ satisfies $h_t(x, y) \leq Ct^{-n/2}$. Consider a function $u \in L^p(M)$. How can we prove that ...
0
votes
0answers
21 views

Trouble understanding holonomic dual basis

I saw a post earlier and it made me a bit confused, so I looked into it and I have some questions about a paper I saw. It said since a tangent vector can be defined $u=d/dλ$ where λ is the parameter ...