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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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.

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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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
16 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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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
19 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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13 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
12 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
11 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
15 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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0answers
8 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
17 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
69 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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0answers
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
44 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
26 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 ...
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2answers
20 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
20 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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15 views

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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28 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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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 ...
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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 ...
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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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15 views

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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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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0answers
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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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 ...
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43 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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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 ...
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2answers
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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
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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 ...
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6answers
98 views

Examples of global properties that don't arise from local knowledge

Let $M$ be a smooth manifold. As an example of a global property that arises from local data we know that if $(M,g)$ is a compact surface without boundary then the Euler characteristic is given by $$ ...
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Curvatres specialized on disc

Consider an open disk of unit radius in the real (two dimensional) plane.If we want to define the Ricci curvature and bisectional curvature on that disc,what will be their equivalent forms and the ...
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Curvature of a planar

Show that the curvature of a curve in 3 dimensions that is parametrized by arclength does not change if we shift the curve or rotate it by a rotation with positive determinant . I tried to prove it ...
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1answer
24 views

Finding the center of mass for a centroid without a convenient symmetry axis

Find the centroid of the lamina described in polar coordinates as $\left \{ \strut \left ( x,y \right )~|~0\leq r\leq 4 \cos\left ( \theta \right ),0\leq \theta \leq \frac{\pi}{3} \right \}$ Having ...
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Harmonic maps between Riemann surfaces

In 'Compact Riemann surfaces' Jost defines harmonic maps between surfaces $S_1,S_2$, with local coordinates z on $S_1$ and metric $\rho^2|du\,d\overline{u}|$ on $S_2$ as $u\in C^2$ solving the ...
2
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2answers
37 views

How to understand conjugate points on a Riemannian manifold?

I'm having trouble grasping what it means for two points to be conjugate on a Riemannian manifold. Could someone provide a geometric or intuitive explanation for this? For clarification: given a ...
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1answer
21 views

How to define gradient of an affine connection

I heard somewhere (and just read on a physics forum) that the gradient of a smooth function $f$ on a manifold $M$ can be defined when $M$ is equipped with an affine connection on its tangent bundle, ...