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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How do conformal maps affect curvature?

Let $(\overline{M}^{n+1}, \langle \cdot, \cdot \rangle)$ be a riemannian manifold with riemanniam connection $\overline{\nabla}$ and consider $M^n \subset \overline{M}$ an orientable hypersurface with ...
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19 views

tensor product of a line bundle with $\bigwedge^k T^*M$

I am reading a source that says: Let $L \to M$ be complex line bundle. Define $\Omega^k(M,L)=C^\infty(M,L \otimes\bigwedge^k T^*M)$. Can someone explain thoroughly what this means? What does it mean ...
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23 views

Proving a curve is spherical iff it satisfies the relation

This question has been asked in a few different way that I have been looking through and trying to understand. On the following answer: Prove that a curve is spherical iff it satisfies the relation I ...
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13 views

Riemann curvature tensor for geometry surfaces on $\mathbb{R}^3$

Let $M\subseteq \mathbb{R}^3$ be a regular surface and $p\in M$. The Riemann curvature tensor is defined by: $$\begin{array}{rcll} R_p:&T_pM\times T_pM\times T_pM&\longrightarrow &T_pM\\ ...
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1answer
12 views

The knowledge of $n=n(s)$ can be used to determine the curvature $k(s)$ and the torsion $\tau (s)$

Question: Show that the knowledge of the vector function $n=n(s)$ of a curve $\alpha:I\rightarrow \mathbb{R^3}$ with nonzero torsion everywhere, determines the curvature $k(s)$ and the torsion ...
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15 views

A question my proof about the line of curvature

I am working on Exercise 2.4.4 of Differential Geometry and Its Application. The problem statement and my work is available at this link. At the end of my proof, I claimed that $ S_p(\alpha') $ is ...
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1answer
15 views

Sum of skew symmetric and symmetric parts of tensors

Denoting the skew-symmetrisation and symmetrisation of a $(0,p)$-tensor $X_{a_1 \ldots a_p}$ by $X_{[a_1 \ldots a_p]}$ and $X_{(a_1 \ldots a_p)}$ respectively, is it always true that $X_{a_1 \ldots ...
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Given the normal versor $n$ to a curve determine its curvature and torsion. [on hold]

Given a curve $\alpha: I \rightarrow \mathbb{R^3} $ and its normal versor $n(s)$ who is known $ \forall s \in I$ and given the Frenet relations \begin{cases}t'=kn\\ n'=-kt-\tau b\\ b'=\tau ...
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18 views

Unit normal vector at inflection point for any curve: Defined or Undefined?

Consider an arbitrary parametric planar (for simplicity) curve: $ \vec{r}(t) = f(t) \,\hat{i} \, + \, g(t) \, \hat{j}$ Differentiable twice over its domain. $ \vec{r'}(t) = f'(t) \,\hat{i} \, + \, ...
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1answer
21 views

Spheres as Symplectic Homogeneous Spaces

Does there exist a description of the odd dimensional spheres as homogeneous spaces of the symplectic group. For $S^7$ it seems to me that we should have $S^7 \simeq Sp(3)/Sp(2)$, but I can't make a ...
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22 views

A specific embedding of semisphere on $R^2$.

I was playing with piece of paper which has the form of semisphere, to be more precise we may assume that it satisfies $x^2+y^2+z^2=1$ for nonnegative $z$. I tried to make it flat without stretching ...
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17 views

moment of inertia and parallel axis theorem

A lamina with density $\delta \left ( x,y \right ) = x^{2}$ has the shape of the disk $\left \{ \left ( x,y \right )|x^{2}+y^{2}\leq 4 \right \}$. Find the moment of inertia of the lamina about the ...
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1answer
22 views

Intuitive understanding into the mean curvature flow

I am trying to develop some intuition into the properties of mean curvature flow of a surface in $\mathbb{R}^3$. As an example, I am trying to understand what happens to a surface of revolution $$S = ...
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16 views

Boundary of a $k-$Cell Definition

In the book I am reading, the Boundary of a $(k+1)-$cell $\varphi$ is defined to be the $k-$chain $$\partial\varphi=\sum_{j=1}^{k+1}(-1)^{j+1}(\varphi\circ \iota^{j,1}-\varphi\circ \iota^{j,0}) $$ ...
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1answer
15 views

True/False? If $\alpha(t)$ is NOT parametrized by arclength, then $T ' · T$ and $T · B$ need not be $0$.

True/False? If $\alpha(t)$ is NOT parametrized by arclength, then $T ' · T$ and $T · B$ need not be $0$. Since $T' = k(s)N(s)$, then $T'·T$ should be zero since $N$ and $T$ are perpendicular. ...
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1answer
12 views

True/False: The derivative of the unit binormal with respect to arclength is always parallel to the unit normal.

True/False: If $\alpha(t)$ is a regular parametrized curve such that $\alpha'(t) \neq 0 $ for any t, then the derivative of the unit binormal with respect to arclength is always parallel to the unit ...
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1answer
17 views

Example of a non-regular curve which has the same geometric image as a regular curve parametrized by arclength

Give an explicit example of: $(a)$ a regular curve parametrized by arclength; $(b)$ a non-regular curve which has the same geometric image as the previous one. Could someone please help me with ...
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11 views

Detailed proof (submersion) : show that the differential is surjective

I'm currently studying manifolds and wanted to have a detailed insight on a part of some proof. This might be very easy, but I can't find the good words to express the correct idea. My definition of ...
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1answer
18 views

Use of implicit function theorem in showing that $f(x) \leq a$ is a submanifold with boundary

This question comes from a statement in John Milnor's "Morse Theory" on page 4. Let $f: M \to \mathbb{R}$ be a smooth function on a manifold $M$. Milnor claims that if $a$ is not a critical value of ...
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3answers
74 views

Distance between two circles on a cube

I found this problem in a book on undergraduate maths in the Soviet Union (http://www.ftpi.umn.edu/shifman/ComradeEinstein.pdf): A circle is inscribed in a face of a cube of side a. Another circle ...
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16 views

careful looking at the linear map dNp with its correspondence matrix

Here, I have some question that remain unsolved for quite a long time. My question is about the differential of the gauss map $dN_{p}$, to start the convenient expression of the symbol and formula, I ...
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18 views

n-1 dimensionnal Hausdorff measure and codimension 1 measure

I've been told that on a n-dimensionnal Riemannian manifold, the Hausdorff measure of dimension n-1 and the codimension 1 measure $v_{-1}$ (defined below) are mutually absolutely continuous. I've ...
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24 views

Geometrical interpretation for curvatures

What is the geometric interpretation for Ricci and Holomorphic Bisectional curvatures in the two dimensional space,like an open ball in the real plane??Any intuitive idea or source will be helpful.
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2answers
46 views

If the Lie algebra is ${\frak g}={\frak a}\oplus{\frak b}$ then the Lie group is $G=AB$?

Let $G$ be a connected Lie group and suppose that its Lie algebra ${\frak g}$ splits into a direct sum of ideals $${\frak g} = {\frak a}\oplus{\frak b}.$$ Let $A$ be the connected Lie subgroup of $G$ ...
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1answer
22 views

How could someone conclude $\check{H}^i (M, \mathbb{R}) = 0$ for arbitrary $M$?

sorry if this is a very stupid question and I'm missing something very trivial, though I could not solve it after thinking for a while. In page 18-19 of ...
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1answer
31 views

Parametrising a curve using curvature and torsion functions

I am trying to get a parametrization of the curve whose curvature and torsion functions are given as $$\kappa(s)= \dfrac{1}{1+s^2} ,\;\; \tau(s) = \dfrac{s}{1+s^2}$$ I know that in general it is ...
2
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2answers
23 views

The configuration space of directed great circles on the sphere

I am reading a proof of Pu's inequality in Katz' book "Systolic geometry and topology". In section 6.4, he describes a fibration $q : SO(3) \to S^2$ which I can't quite figure out. First, he thinks ...
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1answer
21 views

Is the global sections functor on smooth manifolds an embedding?

Is the functor $\Gamma:M \mapsto C^{\infty}(M)$ an embedding from the category of smooth manifolds to the (opposite) category of real algebras? Or equivalently, one has a map of sets $C^{\infty}(M,N) ...
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How to define and compute the norm of a vector with riemannian metric?

Let us consider for example, the riemannian metric $g=e^xdx^2+dy^2$ (it is symmetric and definite positive), with associated matrix $\begin{pmatrix} e^x & 0\\ 0 & 1 \end{pmatrix}$. Consider ...
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Constructing the Hodge Laplacian from the Laplace-Beltrami one

I know, in principle, how to construct the Hodge Laplacian with the aid of the Hodge star. Is it possible, though, to construct it only from the Laplace-Beltrami operator? I have already found a way, ...
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29 views

How to calculate the length of this plane curve (loxodrome/rhumb line)?

I am trying to calculate the length of a (what I believe is) a loxodrome, using differential geometry. I am given a curve $\gamma(t)=\big(\theta(t),\varphi(t)\big)\subset \mathbb S^2$ that ...
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1answer
28 views

What is the meaning of “infinitesimal structure”?

Reading a Differential Geometry book I found this sentence: "A main theme in analysis on metric spaces is understanding the infinitesimal structure of a metric space." I cannot understand the meaning ...
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23 views

Finding the chern classes of line bundles over projective space using homotopy classes of clutching functions

I have just started to learn about characteristic classes and before learning more about the ways to compute them it would be nice to compute some examples using tools I already know. I only started ...
3
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1answer
6 views

Extrinsic Curvature of Surface of Codimension > 1

We can define the extrinsic curvature of a codimension-one surface as $$K_{ab} = q_a^{\phantom{a}c} q_b^{\phantom{b}d} \nabla_c n_d,$$ where $n^d$ is the normal vector to the hypersurface and ...
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Mean and Gaussian curvature - normalization to interval $[0, 1]$ [on hold]

I can compute curvature of the $2.5D$ surface. Problem is, I need the results scaled in interval $[0,1]$ (or $[-1,1]$). Is it possible to compute this directly or I need to compute all curvatures of ...
2
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1answer
25 views

The Levi-Civita Connection on the Hyperbolic Plane

In this question here, I asked about computing the Levi-Civita connection matrix on the Hyperbolic Plane, defined as $\mathbb{H}^2=\{x+iy\in\mathbb{C}\ \vert\ y>0\}$ equipped with the metric $g = ...
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1answer
28 views

Does stereographic projection preserve or reverse orientation?

Let $S^n\subset\mathbb{R}^{n+1}$ denote the standard unit sphere with normal bundle $\nu$, let $N=(0,\dots,0,1)$ and $S=(0,\dots,0,-1)$. Then there are two sterographic projections $$\sigma_+\colon ...
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finding a tubular neighborhood

Assume $f \colon M \to N$ is a smooth mapping between smooth manifolds. Assume $U \subset M$ and $V \subset N$ are submanifolds. Assume $f(U) \subset V$ and $(d_xf)^{-1}\left(T_{f(x)}V\right) = T_xU$. ...
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0answers
11 views

Are involutes always regular?

Suppose the curve $\alpha:I \rightarrow \mathbb{R}^2$ is an involute of a regular curve. Does $\alpha'(t)\neq 0$ hold for all $t\in I$?
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1answer
31 views

Linear Connection on the Hyperbolic Plane

For the upper half-plane $\mathbb{H}^2=\{x+iy\in\mathbb{C}\ \vert\ y>0\}$ equipped with the metric $g = \frac{1}{y^2}(dx^2+dy^2)$, I computed the Christoffel symbols as follows: ...
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23 views

Sectional Curvature, Gauss curvature

I have a problem with a computation which shows that the sectional curvature coincide with the Gauss Curvature in dimension 2. This is the definition of sectional curvature I am using: ...
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1answer
39 views

Rulings of One Sheet Hyperboloid

Let $M$ be a hyperboloid of one sheet satisfying $x^2+y^2-z^2=1$. Show that $x(u,v)=(\frac{uv+1}{uv-1},\frac{u-v}{uv-1},\frac{u+v}{uv-1})$ gives a parametrization of $M$ where both sets of parameter ...
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0answers
28 views

some question about the Jacobian of the differential of the Gauss Map $dN_p$

Here, I have some question that remain unsolved for quite a long time. My question is about the differential of the gauss map $dN_{p}$, to start the convenient expression of the symbol and formula, I ...
0
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3answers
48 views

How to calculate $f(x, y, z)$ given $d f = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy +\frac{\partial f}{\partial z} dz$

On a manifold with local coordinates $(x_1, \ldots, x_n)$ I have a closed 1-form $\omega$ for which $d \omega = 0$ holds. This means There must be a function $f(x_1, \ldots x_n)$ for which $d f = ...
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2answers
42 views

Is the Lie bracket of two vector fields well defined?

I want to understand what exactly means to ask the question if the Lie bracket $[X,Y]$ of two vector fields $X,Y\in \mathcal{XM}$, where $\mathcal{M}$ is a differentiable manifold, is well defined. ...
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0answers
18 views

Exponential map and $\lim_{v,w \to 0} \frac{d(\exp(v),\exp(w))}{\|w-v\|}$

Let $v,w \in T_{p}M$. Prove that $$\lim_{v,w \to 0} \frac{d(\exp(v),\exp(w))}{\|w-v\|}=1$$ I completely don't know how to start. Thanks for any hint. It is an exercise to lecture based on ...
2
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0answers
44 views

Holonomic splitting

I am reading the book "Introduction to the h-Principle" by Eliashberg and Mishachev. At the moment I try to understand the Section 1.7 Holonomic splitting on page 12 but without success. I do not ...
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0answers
16 views

Singularities of composite function

Given a smooth, compact manifold $M$ (of dimension much less than $n$) and two maps $f:\mathbb{R}^n \rightarrow M$, $g:M\rightarrow \mathbb{R}$, I want to understand the topology of the critical set ...
4
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2answers
49 views

What can be said about the leaves of a regular foliation?

I was wondering about the following. Let $M$ be a (smooth, closed, connected and oriented) manifold endowed with a regular foliation (i.e. such that all the leaves are smooth submanifolds of the same ...
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0answers
30 views

Confused in some basic concept about the differential of the Gauss Map $dN_p$ [duplicate]

Here, I have some question that remain unsolved for quite a long time. My question is about the differential of the gauss map $dN_{p}$, to start the convenient expression of the symbol and formula, I ...