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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Why Can't we define the differentiation of vector fields in the same way as in $\mathbb{R^{n}}$

In $\mathbb{R^{n}}$, if $X$ is a vector field on $\mathbb{R^{n}}$, and $X=$$\sum_{i=1}^{i=n}$ $X^{i}$ $\frac{\partial}{\partial x^{i}}$, $X^{i}$ $\in$$C^{\infty}(p)$. Then 1.The differentiation of ...
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71 views

Normal Bundle of Twistor lines

I am reading the paper "hyperkaehler metrics and supersymmetry" by Hitchin etc.. Here is the link: ...
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1answer
473 views

Calculate Angular Position from a gyro

I am working with a gyro that outputs Angular Velocity. I am wondering how I would calculate Angular Position from this data. I have read a couple different places that you need the (sample rate, ...
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157 views

Circle tangent bundle over $S^{2}$

Let $S_{r}^{2}$ be a sphere of radius $r$ and let $TS_{r}^{2}$ be its tangent bundle. If $SS^{2} = \{(x,p)\in TS_{r}^{2}| |p|=1 \}$ be the circle tangent bundle of non zero radius . Then are there ...
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505 views

push forward of vector field

In Gauge Fields, Knots, and Gravity, exercise 18 is the following: Show that if $\phi:M \to N$ we can push forward a vector field $v$ on $M$ to obtain a vector field $\phi_*$ on $N$ satisfying ...
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73 views

Induced sequence of global sections

I'm reading Differential Analysis on Complex Manifolds by Raymond O. Wells. It states the following in the beginning of section 3 of chapter 2 on page 51: Consider a short exact sequence of sheaves: ...
3
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1answer
114 views

Topology of manifolds

Where can I find a stricter presentation of topology of manifolds, then in section 0.4 in Griffiths-Harris? For example, they define the map $H_k \times H_{n-k}$ by presenting a cycle by a submanifold ...
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206 views

Which coefficients of the characteristic polynomial of the shape operator are isometric invariants?

Let $M^n \subset \mathbb{R}^{n+1}$ be an isometrically immersed Riemannian hypersurface. The shape operator $s$ is the $(1,1)$ tensor field characterized by $$\langle X, sY \rangle = \langle ...
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64 views

example of an one form from a $C^{\infty}$ function

For any $C^{\infty}$ function $U\rightarrow\mathbb{R}$ we can construct a $1$ form $df$ called a differential of $f$ as follows $p\in U$ and $X_p\in T_p(U)$ we define $$(df)_p(X_p)=X_p f$$ I want an ...
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2answers
156 views

How is differential form different from ordinary calculus objects?

I am going to learn differential form soon, but after reading some introductory parts of my texts, I couldn't get why differential form is needed and how it is different from ordinary mathematics ...
7
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386 views

A Cover of an Orientable Manifold is Orientable

The following question comes from Introduction to Smooth Manifolds by Lee: Suppose $\widetilde{M}$ smoothly covers $M$ where $M$ is orientable. Show that $\widetilde{M}$ is orientable. I think the ...
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123 views

Smooth maps between Euclidean spaces

There is a question that has bothered me for quite a long time. When we define smoothness of a function(or a map from one space to another), we define it as "has continuous partial derivatives of all ...
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1answer
121 views

Lie bracket of coordinate frame

Suppose I have a local nbd $U$ in a manifold centered around $p$ and a chart on it. The chart is given by $X=(x_1,x_2,\cdots,x_n):U\rightarrow V\subset\mathbb{R}^n$. I consider the vector fields ...
2
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0answers
52 views

Dual connections, bracket

If $\nabla$ is a torsionfree connection and $(\nabla_{X}J)Y=(\nabla_{Y}J)X$, J- an almost complex structure, and $\nabla_{X}^{*}Y:=J\nabla_{X}(JY)$ its dual connecion. Is it correct to conclude that ...
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76 views

Dimension of intersection of two manifold

For any $f\in C^\infty(X)$, $X$ smooth manifold. Define $$X_{df}:=\{(x,df_x): x\in X, df_x= T^*_x X\}$$ $$X_0:=\{(x,\zeta): \zeta=0 \text{ in } T_x^*X\}$$ In the exercise we are asked for proof: If ...
2
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1answer
155 views

Does a 3-Dimensional coordinate transformation exist such that its scale factors are equal?

Let $\vec r=(x,y,z) $ be the position vector expressed in Cartesian coordinates. Let us define the coordinate transformation as $\vec r(u,v,w)=(x(u,v,w),y(u,v,w),z(u,v,w)) $ The scale factors are ...
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331 views

When does a vector field admit orthogonal fields?

My question is: Let $\,X$ be a nonvanishing smooth vector field over an open subset $U \subset \mathbb{R}^3$. Which conditions on $X$ guarantee the existence of a smooth nonvanishing vector field ...
3
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487 views

Geodesics on a 2-sphere

I've been doing some work where I need to find the geodesics in a given Riemannian Manifold. Let's take the example of the two sphere, for simplicity, with unitary radius. The distance between two ...
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33 views

Nearly Kaehler and special Kaehler manifolds

We know that the most important example of a nearly Kaehler manifold is the sphere $S^{6}$ and that $(\nabla_{X}J)Y=-(\nabla_{Y}J)X$ is valid in this case (J - an almost complex structure). Similar ...
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806 views

Are Clifford algebras and differential forms equivalent frameworks for differential geometry?

I recently discovered Clifford's geometric algebra and its application to differential geometry. Some claim that this conceptual framework subsumes and generalizes the more traditional approach based ...
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400 views

Gaussian curvature

Let us take a surface $z =x^3-3x^2+y^2$. If you calculate the critical points on the surface it will come as $(0,0)$ and $(2,0)$. Moreover one can find the local behavior of the critical points. It ...
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339 views

How to derive covariant derivative and Lie derivative of tensors

1) As title says, how does one derive the following equation for covariant derivate of tensor: $A^{\alpha}{}_{;\beta} = A^{\alpha}{}_{,\beta} + \Gamma^{\alpha} {}_{\gamma\beta}A^\gamma$ where ...
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2answers
266 views

How to show something is a contraction?

If we let $X$ be a complete metric space, and let $S:X\to X$ be a map, such that $S^m$ is a contraction. We now want to show, that $S$ has a unique fixed point This is what I've thought so far: Due ...
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How to find the area vector of a shape?

If I have a simple 3-d shape like a square plate connected to an identical square plate at one edge and they are at an angle of 90 degrees, how would I find an area vector that describes it? I think ...
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331 views

Isometric but differently shaped surfaces in $\mathbb{R}^3$

We have the following chain of inclusions for surfaces in $\mathbb{R}^3$ $M_1,M_2$:      $M_1,M_2$ have the same shape, i.e. are related by an ambient isometry ⇆ ...
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Different notions of isometry for Riemannian 2-manifolds

There are two notions of isometry between Riemannian 2-manifolds: a distance-preserving map $f$ with $d(x,y) = d(f(x),f(y))$ and a "metric-preserving" map $f$ with $I(x) = I(f(x))$ ($I(x)$ being ...
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1answer
179 views

Every point in S is umbilical $\rightarrow $ S is a plane or sphere.

Umbilic points on a connected smooth surface problem Here we have a proof that if every point in a surface $S\subset\mathbb{R}^3$ is umbilical then it is contained in a sphere or a plane. But this ...
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1answer
161 views

Lie derivative of curvature

Let $M$ be a Kähler manifold, with Kähler metric $g$. Let $X$ be a holomorphic Killing vector field of $g$, i.e. $\mathcal{L}_{X} g = 0$, where $\mathcal{L}_{X}$ is the Lie derivative along $X$. Let ...
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1answer
95 views

question about conformal map

Thank you for let me ask question I am really enjoy with this website. It is great website I have question about geometry for expert geometry what is the definition of conformal map and the condition? ...
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3answers
596 views

reconstructing space curves from curvature and torsion

Given $\kappa (s)$ and $\tau (s)$ and a frenet apparatus $\lbrace T_0,N_0,B_0 \rbrace$, how can you reconstruct a space curve? I know I need to use the frenet-serret equations, but I can't put a ...
2
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1answer
90 views

Cohomology with Coefficients in the sheaf of distributions

It just occurred to me that one could form the sheaf of distributions $F$ on any manifold where for an open set $U$ we have $F(U)$ is the algebra of distributions on $U.$ What does cohomology with ...
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prove that is not conformal map

I have fun with this website. It is very helpful and great. I have a question in geometry. I try to solve it but it needs imagination to figure out the point on the surface. This question makes me ...
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0answers
33 views

Dual connections, examples

If $\overline\nabla^{*} $ is a dual connection of connection $\overline\nabla $, and we have the Gauss equations: \begin{align} \notag \overline\nabla^{*}_{X}Y=\nabla^{*}_{ X}Y + h^{*}(X,Y) ...
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162 views

Cech Cohomology Of Pullback Linebundle

my question is as follows. Let $\chi$ a compact Calabi-Yau 3-fold and $A,B \subset \chi$ two 2-complex dimensional manifolds such that their intersection $C := A \cap B$ is a 1-complex dimensional ...
3
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1answer
209 views

Maximum principle for minimal hypersurfaces

The well known local version of the maximum principle for minimal hypersurfaces asserts that if two minimal hypersurfaces $ M_1 $ and $ M_2 $ of $ R^n $ has a common point $ x_0 \in M_1 \cap M_2 $ ...
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136 views

First fundamental form to find arc length and angle

If the first fundamental form of a surface is $I = du^2 + (u^2+a^2)dv^2$, find the arc length of each edge and each angle of the triangle enclosed by the curves C_1: u = (a/2)v2, C2: u= (-a/2)v2, C3: ...
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1answer
198 views

A question about laplacian of the second fundamental form

Let $ f:M \rightarrow N $ be an immersed oriented hypersurface, $ e_{1}, \ldots e_{n},e_{n+1} $ be an orthonormal frame of $ N $ such that $ e_{1} \ldots e_{n} $ is an orthonormal frame of $ M $. Let ...
3
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0answers
214 views

Solving systems PDE's

I have a bit of trouble solving a system of first order PDE's, that I get by solving a boundary issue problem in gravitation (here). I have six equations: ...
4
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2answers
154 views

two points of the longest distance in a convex curve will be the vertices?

First we give the necessary definition: A vertex of a curve $\gamma(r)$ in $\mathbb{R}^2$ is a point where its signed curvature $\boldsymbol{k}_{s}$ has a stationary point, i.e.,where ...
4
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1answer
158 views

Deriving equations for the “Bianchi-Pinkall torus”

I am trying to work out explicit parametric equations for the "Bianchi-Pinkall flat torus" as depicted in this note, but I seem to have gotten stuck in understanding the descriptions given in that ...
2
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1answer
146 views

How to construct pseudospherical surfaces from sine-Gordon solutions?

Due to my not being very skilled in differential geometry, I want to ask if there is a reference (book, paper, etc.) that explicitly works out how one constructs the parametric equations of a ...
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34 views

Boundaries- regularity and local parametrization

Suppose we have a bounded domain $\Omega \subset \mathbb{R}^3$ with $C^2$ boundary.Let $x_0 \in \partial \Omega$. We choose a $X_1,x_2,x_3$- coordinate system such that the $x_1,x_2$-plane is ...
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2answers
71 views

Finiteness of fixed points of a Lie group action

Let $\psi: G\rightarrow \mathrm{Diff}(M)$ be a smooth non-trivial action of a compact connected Lie group $G$ on a compact connected smooth manifold $M$. Under which assumptions there will be a ...
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60 views

Find point $X$ such that line through plane $E$ and sphere $S$ meet at $(0,0,1)$ (stereographic projection)

Find the point $X$ such that the line going through the plane $E$ and sphere $S$ meet at the point $(0,0,1)$ (stereographic projection). Let $S$ denote the unit sphere $$S = \{(x,y,z) \in ...
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1answer
335 views

Geometric proof of the geodesics of a sphere?

I have seen the standard variational proof that great circles are the geodesics on the $2$-sphere. Do you know a purely geometric proof of this fact, not involving calculus of variations or ...
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1answer
326 views

differential geometry about neither local isometry nor conformal map

hi I need help in geometry I try to get the the idea but I couldn't solve so. It becomes great if someone try solve with explain to me by using figures if possible. I see this question change a lot of ...
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0answers
24 views

density of $\mathcal{C}_1$ surface in a point

Let us have an $A \subset \mathbb{R^d}$, that is a $k$-dimensional $\mathcal{C}_1$ surface (obviously $k<d$) and let $a \in A$. Why then is $\Theta^k(A,a)=1?$ Of course $\Theta^k := \lim_{r ...
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1answer
117 views

Conjugation of two circle diffeomorphisms

I'm not very familiar with the dynamics of circle maps, and incidentally I realized that an answer to a question concerning circle diffeomorphisms can help me to solve a problem related to a discrete ...
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2answers
34 views

Differentiation along surface

I have this question I got when trying to solve a physics problem and I don't know which topic it belongs to. Please redirect me if anyone asked the same question before. I have a function ...
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160 views

Heat Kernel Asymptotics on Manifold with Boundary

On a closed Riemannian manifold $M$, the heat kernel $k_t(x, y)$ of the Laplace-Beltrami operator (or more general of any generalized symmetric Laplace-type operator acting on sections of a vector ...