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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computing transition function of tangent bundle $S^n$

I'm just starting to learn about vector bundles, I want to compute the transition functions of the bundle $TS^n$. I started with the stereographic atlas $U_1 = S^n - \{N\}$ and $U_2 = S^n - \{S\}$ ...
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26 views

Under what condition on f is this parametrized curve regular?

Consider a parametrized curve in $\mathbb R^2$ given by $$ \gamma (t)=(f(t)\cos(t), f(t)\sin(t)) $$ where $f$ is a smooth function of $t$. Under what condition on $f$ is $\gamma$ regular? I took the ...
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Why does the arc-length formula have form $\int_a^b\left|\left|\frac{d\vec{f}(t)}{dt}\right|\right|_2dt$ for C1 curves?

This discussion focuses on $\mathcal{C}^1$ curve on $\mathbb{R}^n$. But feel free to talk about the case where we only have a continuous curve or the scenario with a manifold with a metric in general. ...
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Transitivity of smooth submanifolds

I was reading through Guillemin and Pollack and was having trouble verifying this for myself. Given $M \subset N$ and $N \subset P$, where $M$ is a submanifold of $N$, and $N$ a submanifold of $P$, ...
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Question surface

We consider the surface $S$ that is defined as the graph of the function $z=2x^2-y^2, x,y \in \mathbb{R}$ Find a basis of the tangent plane $T$ of the surface $S$ at the point $M=(-1,2,-1)$ Find a ...
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30 views

Relation between $\text{Hom}_{\mathsf{Alg}_{\mathbb{R}}}(\mathcal{C}^\infty(M),A) $ and $ X \otimes_\mathbb{R} A$?

This question is a little bit of a shot in the dark, but maybe someone stumbled over it before... Let $M$ be a (simply connected) smooth manifold modelled on a locally convex space $X$ over ...
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48 views

How to organize my learning in Maths?

I m working on a problem in mechanics of material which concerns about the variation of shapes. I need to understand the deformation of material. I m a civil engineering graduate. All my understanding ...
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49 views

Differential and Integral calculus.

Can anyone here explain me, why do we take the Centre of mass of a conical shell using slant height and $dl$ whereas the centre of mass of a solid cone is calculated using the vertical height and ...
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1answer
28 views

Hausdorf Dimension of a manifold of dimension n?

Let's say that $M$ is a differentiable manifold of dimension $n$. (This includes that $M$ is nonempty and second countable, so that it can be embedded into some Euclidean space, and is thus ...
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Problem 1, Chapter 3 - A Comprehensive Introduction to Differential Geometry - Spivak

There seems to be an error in item (d) of Problem 1, Chapter 3 of Spivak's A Comprehensive Introduction to Differential Geometry, 3rd Edition, which I transcribed below. There is something wrong with ...
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33 views

Existence of a smooth curve with specific properties

Let $\gamma: [0,T] \to \mathbb R^n$ be a differentiable curve with the property that for any $t_0 < t_1$ and any vector $v \in \mathbb R^n$ $$\langle \gamma(t), v \rangle \ne 0, \;\;\text{ for all ...
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Einstein Summation - does the following equality hold: $a_{ij} x_i y_j = a_{ij} y_i x_j$

Does equality hold when $x_i = y_i$ and $x_j=y_j,$ and $ i, j = 1, ..., n $.
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2answers
31 views

Einstein Summation: How do I show $a_{ij} (x_i + y_j) \not= a_{ij}x_i + a_{ij}y_j $?

Einstein Summation: How do I show $a_{ij} (x_i + y_j) \not= a_{ij}x_i + a_{ij}y_j $?
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1answer
46 views

Proof of curvature of a curve described by Polar Coordinates

I have been looking everywhere for a proof on the curvature of a plane curve that is represented in polar coordinates. I am close in proving it myself, however, I seem to be missing a particular part ...
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39 views

which surfaces have (for a large area) a constant negative curvature?

There is no surface in $ R^3 $ that can represent the complete hyperbolic plane (Hilberts theorem) so we always have to do with a surface that is not completely equivalent, has a cusp somewhere, but ...
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19 views

Find real solution for an inhomogene system

I have an inhomogene differential equation system $\begin{pmatrix}\dot{x}_1 \\ \dot{x}_2\end{pmatrix} = \begin{pmatrix}-1 & 3 \\ -3 & -1\end{pmatrix} \begin{pmatrix}x_1 \\ x_2\end{pmatrix} + ...
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48 views

Interpretation of differential form

We know what is the interpretation of a total differential, ex.: $$df=\frac{\partial f}{\partial x} dx+ \frac{\partial f}{\partial y} dy$$ But what is the interpretation of a 1-form and its exterior ...
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Question about integrating differential forms

Maybe it's stupid question, by why: $$\int_S Fdx\wedge dy=\int_S Fdxdy$$ And is calculating a surface integral $$\int_S Fdx\wedge dy+Gdy \wedge dz+H dz\wedge dx=\int_S Fdxdy+\int_SGdydz + ...
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54 views

The projections are differentiable $1$-forms.

Suppose $M$ is a surface and suppose $X: U \subset \mathbb R^2\rightarrow M$ is a coordinate patch. Then for every $p \in X(U)$, the pair of vectors $(X_u(X^{-1}(p),X_v(X^{-1}(p) )$ is a basis of the ...
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52 views

Space of oriented lines in $\mathbb{R}^{n+1}$ as symplectic quotient.

I've been working out a nice example of symplectic reduction, and have come to a solution only after quite a lot of effort. So I was wondering if anyone knew a more straightforward route to the ...
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29 views

Curvature proof of a convex plane curve

Having a little trouble with part b. Is there a way to show that this curve would be arc length paramaterized? I am assuming that we cannot say this. If it is not we can take alpha', alpha'' and ...
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60 views

A smooth function $f:S^1\times S^1\to \mathbb R$ must have more than two critical points.

I am trying to show that a smooth function $f:S^1\times S^1\to \mathbb R$ must have more than two critical points. Since $f$ attains maximum and minimum, it must have at least two critical points. ...
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30 views

Connected and simply connected neighborhoods

Suppose that $E \to M$ is a (smooth) vector bundle over smooth manifold $M$. One can find the covering $\{U_i\}_i$ with the property that $E|_{U_i}$ is trivial vector bundle. The prooblem is the ...
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24 views

Curve in union of hyperplanes

If a smooth curve $\gamma: [0,T] \to \mathbb R^n$ is contained in the union of hyperplanes $$ \bigcup_{i=1, \dots, N} H_i$$ does it then follow that one can always find time intervals $[t_0, t_1]$ ...
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Tangent space of loop space

In this article (section 2.1) it is written For $\gamma \in LM$, the formal tangent space is $\Gamma(\gamma^*TM)$, the space of smooth sections of the pullback bundle $\gamma^*TM \longrightarrow ...
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Pullback and quadratic differentials on Riemann surfaces

Suppose we have some meromorphic quadratic differential $q=\phi\left(x\right)\mathrm{d}x^2$ on a punctured Riemann surface $\mathcal{C}$, and that $q$ is the pullback via a function $f$ of a ...
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36 views

Prove that the line is tangent to the curve at the point.

Hello can someone please walk me through part a and b of the below question? I really want to understand it but am having a hard time figuring out the solution. I know how to calculate curvature for a ...
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1answer
20 views

Area of pseudospherical segment

Surface area of segment of a sphere radius $a$ at the equator, between two parallels, is given by $ 2 \pi a (z_2-z_1) $,where $z_2, z_1$ are heights of spherical segment at radii of parallel circles ...
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1answer
57 views

Why Riemannian metrics have to be smooth?

Why do Riemannian metrics have to be smooth? Can you give an example of a smooth curve with a none smooth metric and show me what possibly will go wrong if our metric is not smooth?
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59 views

What is global differential geometry?

What is the difference between local and global differential geometry? I cannot find their (exact) definitions. There are some other terms in geometry like "rigid" (e.g. that structure is more rigid ...
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41 views

Exercise 5.8 from Lie Group, Daniel Bump

In the exercise 5.8 Bump has asked to prove that the group $Sp(4)$ over complex numbers, which is usual complex embedding $U(4)\cap Sp(4,\mathbb{C})$, can be described by, $$\left\{\begin{pmatrix} ...
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120 views

Why is differential geometry called differential geometry?

Why is differential geometry called differential geometry? Why it is not called differential and integral geometry? Isn't integration and finding areas as important as differentiation? Is it the case ...
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23 views

Tangent Spaces of Smooth Manifolds [on hold]

If $\phi : M \rightarrow M'$ and $\psi:M \rightarrow M'$ are differentiable function which agree on some neighbourhood of m, show that $\phi_{*,m}=\psi_{*,m}$. Where $\phi_{*,m}$ and $\psi_{*,m}$ are ...
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34 views

Does the cup product on de Rham cohomology induce a nondegenerate bilinear form?

I have small issue I came across in the following. Suppose $M$ is a compact, oriented manifold of dimension $4n+2$. I want to prove that the de Rham cohomology group $H^{2n+1}(M)$ are even ...
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1answer
31 views

The Uniqueness Part of the Smooth-Manifold-Chart-Lemma in John M. Lee's Introduction to Smooth Manifolds.

I am trying to understand the proof of Lemma 1.35 (Smooth Manifold Chart Lemma) of John. M. Lee's Introduction to Smooth Manifolds, 2nd Edition. The Lemma is an existence-and-uniqueness-lemma. I ...
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Solutions to do Carmo's Riemannian Geometry [on hold]

I have lot difficult in solving problem in Riemannian Geometry by Manfredo do Carmo. Does anyone know solution book of those? I just want ask if anybody know so! Gracias!
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Pullback of a form under the retraction $r\colon \mathbb{R}^n\setminus\{0\}\to S^{n-1}$.

The following is from Spivak's DG Lemma 7 in Chapter 8, but I'm muddled in a computation. Define two $(n-1)$-forms on $\mathbb{R}^n\setminus\{0\}$ by $$ ...
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2answers
26 views

Compatible intersecting coordinate patches must map to the same dimensional $\mathbb R^n$.

I have a problem where I have to show that for two intersecting open subsets $U$ and $V$ of a topological manifold, if we have two homeomorphisms $\phi : U \to \mathbb R^n$ and $\psi: V \to \mathbb R ...
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1answer
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Give a closed plane curve C with k (curvature) > 0 that is not convex, Draw closed plane curves with rotation indices 0, 2, -2, and 3

1.) Give a closed plane curve C with k (curvature) > 0 that is not convex. can someone please explain these concepts to me. How can you have a closed plane curve like this? Do you used signed ...
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1answer
47 views

What is a squashed 3-sphere?

I have found the term "squashed 3-sphere" used in the literature but could not locate a precise definition of it. I suppose it is topologically a 3-sphere with a metric different from the round one. ...
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1answer
27 views

“Hyperboloid like surface” as hyperbolic plane / pseudosphere

A pseudosphere is an surface wth a constant negative curvature. In most publications, it is almost given that the tracioid (rotated tractrix) is the surface that has a constant negative curvature, ...
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22 views

On Steiner Surface

I would be very grateful if you help me with explicitly proving that the Steiner surface is a topological manifold. Thanks in advance!
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1answer
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inverse of stiefel-whitney class of product bundles

Let $\xi_1,\cdots,\xi_n$ be vector bundles over $M$. Let $\omega$ be the Stiefel-Whitney class and $\bar\omega$ be the inverse. By an exercise in Milnor's book, $$ w(\Pi_{j=1}^a ...
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A Couple Formulas in Besse's “Einstein Manifolds”

In Besse's "Einstein Manifolds," Chapter 6D, there are 2 formulas which I am interested in, which apply to compact Riemannian $4$-manifolds: ...
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26 views

The signature of a product of surfaces

If $\Sigma_1$ and $\Sigma_2$ are surfaces (i.e. compact, oriented 2-manifolds without boundary), is the signature $\tau (\Sigma_1 \times \Sigma_2)$ well-known? Recall that the signature is the ...
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28 views

Gluing oriented manifold along boundaries

Let $M_1$ and $M_2$ be oriented manifolds with boundaries. Suppose they have homeomorphic boundaries. I want to glue $M_1$ and $M_2$ along the boundaries via some homeomorphism. To ensure that the ...
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1answer
87 views

Shape operator of the sphere.

I want to compute the Weingarten operator (shape) for the sphere $\{(x,y,z) \in \mathbb{R}^3 \ : \ x^2 + y^2 + z^2 = 1\}$. I am given the adapted frame: $$\left\{\begin{array}{l} E_1 = \cos \varphi ...
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26 views

$C^0$ estimate for solutions of the Neumann problem

I am interested in a reference for (or counterexample to!) a particular $C^0$ estimate for solutions of the Laplace equation with Neumann boundary conditions. More precisely, let $(M,g)$ be a smooth, ...
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36 views

normal vectors in spaces where $n > 3$

I am reading Lovelock and Rund's book on Tensors and they make a statement that I wanted to validate about normal vectors in high-dimensional spaces. It should be remarked that the above ...
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Quaternion Calculus

I was reading a note on Quaternion(Link) and I am happened to read a section regarding a solution of quaternion differential equation. I am putting that segment as picture format here for more ...