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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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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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
11 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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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
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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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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
19 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
17 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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19 views

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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25 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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27 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
22 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
27 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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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
29 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
37 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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37 views

Hausdorff quotient space

consider a smooth manifold $M$ and a group action, i.e. a group homomorphism $\phi: G\rightarrow S(M)$, where $S(M)$ denotes the group of diffeomorphisms of $M$. Suppose that for all $K\subset M$ ...
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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 ...
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3answers
46 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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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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38 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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14 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
45 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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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 ...
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6answers
93 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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25 views

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 ...
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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, ...
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0answers
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Compute the degree of $\phi:(x_1,x_2,x_3,x_4) \mapsto (x_1,-x_3,-x_2,x_4)$. Does $\phi$ preserve orientation?

Consider the map $\phi:S^3 \rightarrow S^3$ given by $\phi(x_1,x_2,x_3,x_4)=(x_1,-x_3,-x_2,x_4)$. Compute the degree of $\phi$. Does $\phi$ preserve orientation? First I want to point ...
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1answer
17 views

Local Representation of Euclidean Connection

I'm trying to understand how connections are locally represented, and the definition I have to work with is this: Let $(x^1,\dots,x^n)$ be local coordinates defined in some chart $U \subset M$ ...
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0answers
19 views

Which of these plane curves are an immersion?

The question asks, which one of these plane curves is an immersion. I'm just checking that I'm correct. A is an immersion because the derivative is everywhere nonzero (thus the derivative is ...
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1answer
15 views

Possibility of regular surface with specific first and second fundamental form matrices

I have met this in diff. geometry class which states: We are to determine if there exists a regular surface in $ R^3 $, $ S = f(u,v) $ with fundamental forms as follows: $ I = \begin{bmatrix} ...
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1answer
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Derivative of an integral on a level set

Consider a mapping $\xi:\mathbb{R}^d\rightarrow\mathbb{R}^k$ such that $D\xi \, D\xi^T>\delta\, I_k$. Here $D\xi:\mathbb{R}^d\rightarrow \mathbb{R}^{k\times k}$ is the Jacobian. Consider a ...
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1answer
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Where is my mistake in this exterior derivative of a $2$-form not being a tensor?

Using the invariant formula for the exterior derivative, one gets that for a $3$-form $\omega$ its derivative is evaluated on vector fields according to $$3 \ {\rm d} \omega (X, Y, Z) = X \ \omega ...
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33 views

Lifting of a Diffeomorphism to an Orientable Double Cover

Let $M$ be a non-orientable smooth $d$-dimensional manifold. Let $\tilde{M}$ be the oriented double cover for $M$; and let $f\in Diff^{1+\beta}(M)$. Define $\tilde{f}:\tilde{M}\rightarrow \tilde{M}$ ...
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Differential geometry [on hold]

If we have integrable distribution D of rank k on a manifold, and we have k functions which are constants on the associated integral manifolds. can we glue together these functions to obtain global ...
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1answer
43 views

How to compute $[\dot c, X]$ on a manifold?

Consider a smooth curve $c : [0,1] \to M$ and $X \in \mathcal X (M)$. How can one obtain an explicit formula for $[\dot c, X]$? I know the theoretical approach: for every $t \in [0,1]$ there exist a ...
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How do I find a smooth map from complex Gr(k, n) to real Gr(2k, 2n)?

I am trying to find a smooth bijective map from complex Grassmannian of $k$-dimensional subspaces of $\mathbb{C}^n$ to the Grassmannian $2k$-dimensional subspaces of $\mathbb{R}^{2n}$, but I don't ...
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16 views

Rotating curve sweeping constant negative Gauss curvature surface

A short line segment rotates around unit circle radius $a$ so that latitude equals longitude or, $ v = u $ so the in the neighborhood of "equator" $ (u\approx a) $ Gauss curvature $ K \approx ...
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2answers
32 views

Vanishing Integral of a differential form without using Stokes' Theorem

In $\mathbb{R}^3$ consider following 2-form given by $$\omega = xy \: dx \wedge dy + 2x \: dy \wedge dz + 2y \: dx \wedge dz$$ and $$A = \{(x,y,z) \in \mathbb{R}^3 | x^2+y^2+z^2=1, z\geq 0\}.$$ ...