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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Inverse of the Stereographic projection $\mathbb{CP}^1 \to S^2$

I've some problems with this exercise: Consider the stereographic projection $$ \varphi \colon S^2 \setminus \{ (0,0,1)\} \to \mathbb{CP}^1\setminus \{[1:0]\}$$ $$ (x,y,z) \mapsto [ x+iy:1-z]$$ and ...
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Rectifying linearly independent vector fields

Suppose we are given two vector fi elds $V_1$ and $V_2$ - defined on $R^n$- such that the vectors $V_1(x)$ and $V_2(x)$ are linearly independent for each $x$. Is it possible to find a diff ...
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question on tangent bundle

Let $X$ be a manifold and consider its tangent bundle $T(X)$ and let $p$ be the usual map $T(X) \to X$. Then why is it locally trivial ? i.e why for all $x\in X$ exist open neighborhood $U$ of $x$ ...
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Why do we need an orientable surface for Gauss map?

I'm learning Differential Geometry recently with do Carmo's book. In the book, Gauss map is define as a differentiable map from an orientable surface $\mathcal{S}$ to $S^2$ in such a way that for ...
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Reparametrisation of closed not closed

I would like an example of a closed curve and a reparametrisation of the same curve that is not closed.
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$d+\Gamma$-understanding connections

$d+\Gamma$ is a formula commonly found in textbooks when one reads about connection. To be more precise: if $U$ is an open set over which a given vector bundle $E \to M$ is trivial then a coonection ...
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Retract of a free $\Omega(\mathbb{R})$-module

Can an open subset X of $\Omega(\mathbb{R}^2)$ be an $\Omega(\mathbb{R})$-module retract of some free $\Omega(\mathbb{R})$-module? Here $\Omega(\mathbb{R}^n)$ denotes the usual topology of ...
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Unit circle circumferential length on a Beltrami pseudosphere

What is the perimeter length of a geodesic circle of unit radius of tangential curvature? .. assuming that the circle lies entirely on one side of its cuspidal equator.
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Guessing shape made by Beltrami

If $ z = r^2 f(n \theta ) $ has constant negative Gauss Curvature, find $ f(\theta) $ or an ODE leading to it, when n is an even integer. I lost my earlier derivation involving elliptic integrals. ...
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Frobenius theorem for singular foliations , what are hypotheses impose over an endomorphism $P:TM\to TM$ that it span a singular foliation in M?

Let $M$ a smooth manifold. Given a morphism $P:TM\to TM$, i.e, $\pi\circ P\equiv Id$ and is linear over the fibers. If we suppose that $P^2=P$, then for each $x\in M$, $P_{x}:T_xM\to T_xM$ is a ...
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24 views

General Tensor Assistance

Sorry if this is a stupid question, but it might help me grok things if I can connect from something that's intuitive to me. Consider a transformation from Cartesian coordinates to spherical ones: ...
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42 views

Geometric Interpretation of QFT Scattering Integrals

Let $f\in C^\infty(\mathbb{R}^n,\mathbb{R}^k)$, and $g\in C^\infty(\mathbb{R}^n,\mathbb{R})$, where $k<n$. How do I compute $$\int_{\mathbb{R}^n}\delta^k(f(\mathbf{x}))\cdot ...
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Standard complex structure in contact geometry

In symplectic geometry there is the standard model: $(\mathbb{R}^{2n}, d(\sum_{i=1}^{n}x_{i}dy_{i}))$ with standard almost complex structure $ J_{0}= \begin{pmatrix} 0 & -E_{n} \\ E_{n} & 0 ...
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The second cohomology of total space of the $\mathbb CP^1$ bundle

$X$ is a closed smooth surface with $L$ a complex line bundle on $X$. Consider the $\mathbb CP^1$-bundle $P(L\oplus 1)$, that is the projectivization of the sum of $L$ and the trivial line bundle on ...
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28 views

Rerformulation of a previous question concerning a problem in physics that involves integration of 2-forms over the sphere

In this question the integral proposed in the posting concerns a physical problem that can shortly be described by the following : Let $J$ be a real valued function on the sphere (in fact it is a ...
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If $f$ is an immersion and $g$ is a submersion, then is $g \circ f $ a local diffeomorphism?

I don't think so; the counter example I had in mind was $f : \mathbb{R}^2 \to \mathbb{R}^3 , f(x,y) = (x,y,x)$ and $g:\mathbb{R}^3 \to \mathbb{R}^2, f(x,y,z) =(x-z, y-z)$. Is my example right?
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Show that the arc length, the curvature, and the torsion of a parametrized curve are invariant under rigid motions.

Show that the arc length, the curvature, and the torsion of a parametrized curve are invariant under rigid motions. If $\alpha$ is a parametrized curve and $\beta= \operatorname M(\alpha(s))$ where ...
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geodesic polar coordinate parallel circles

When is it possible to have the same constant geodesic curvature on all parallels of a constant Gauss curvature surface? EDIT: picture added.
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Revolution surfaces of constant Gaussian curvature k=1

Prove that all revolution surfaces $(\phi(v) \cos u ,\phi(v) \sin u,\psi(v)) $ of constant Gaussian curvature $k = 1$ is one of the following types: $\phi(v)=C\cosh v$ and $\psi(v)=\int_0^v ...
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Visualization of Gauss Bonnet geometric objects

Where can we get to see some individual surface/line combinations in isometry visualizations with constant $ \int k_g ds $ (say total tangential rotation) ? Or with constant integral curvature ...
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How To Find a Set of Points Farthest Apart Within 3D Solid

I am trying to find out a method to solve the following problem: There are two parameters: 1) There is a solid 3D region plotted in a cartesian coordinate system. 2) There is a number of points that ...
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30 views

Relationship between differentiation and integration of vector fields?

Let $V\in\Gamma(T\mathbb{R}^n)$ be a vector field and $\gamma:[a,b]\to \mathbb{R}^n$ a curve. Let $\nabla$ be the Euclidean connection, i.e. $\nabla_XY=XY^k\frac{\partial}{\partial x^k}$. We have a ...
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Computing transition map of $S^2$.

First please have a look at the cruddy diagram I have drawn. (it is at angle because my camera casts a shadow if I photograph it from above) Define the coordinate charts that map a portion of the ...
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Orientation of surfaces

From the book: Fixing a parametrization $x(u,v)$ of a neighborhood of a point $p$ of a regular surface $S$, we determine an orientation of the tangent plane $T_p (S)$, namely, the orientation of the ...
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$n$-connectivity of principal $G$-bundle

Page 202 of Switzer's Algebraic Topology: Let $\xi=(E,p,B)$ be a principal $G$-bundle such that $E$ is $n$-connected. Then for any pointed CW-complex $(X,x_0)$ of dimension at most $n$, the ...
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19 views

Geodesic formulation from surface parametrization

What differential relation f (u,v,du/dv)=0 can be used to convert parametrization of a two parameter surface X(u,v) into one parameter geodesics on surface X ?
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“Circle” on pseudosphere

How should parametrization of the 2 parameter surface of a pseudosphere ("latitude" u and longitude v) change to result in a 1 parameter curve of constant geodesic curvature? EDIT: In other ...
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Evolute of a cycloid

Find the evolute of the cycloid: $x = u + \sin(u)$, $y = 1 + \cos(u)$ Hint: given is that $T = \big(\cos (u/2), − \sin(u/2)\big)$. Is it that we differentiate $x$ and $y$ and then make it ...
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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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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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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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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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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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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 ...
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Picard theorem on ODE, question with initial data

How one can prove, that the solutions depend smoothly on the initial data for Picard Theorem on ODE?
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Quaternion Solution of the Rotation Equation

I am trying to make a connection between a 3-d vector ODE with a quaternion ODE and a possible solution in quaternion. In the following, a vector $v$ in $R^3$ is interpreted as the vector part of the ...
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definition of critical point defined in terms of differential map

I am having a problem understanding the definition of a critical point in do carmo's Differential Geometry of Curves and Surfaces. He notes in page 58 that a point $p \in U $ is a critical point of ...
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53 views

use Noether normalization theorem to integrate differential forms over singular subvarieties

Let $X \subset \mathbb C^n$ be an analytic subset. I would like to show that locally around any point $x \in X$ the regular part $X_\text{reg}$ has finite volume, perhaps using the theorem below. ...
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Radial & perimeter growth and Gauss curvature fall

A flat circular patch radius $r$ , initial perimeter $2 \pi r$ and initial patch area $ \pi r^2$ grows (dilates) or shrinks, non-isometrically. Shrinkage is associated with radial and perimeter ...
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19 views

Maximize first co-ordinate on general ellipsoid

I have an ellipsoid of the form x^TAx=k , where A is 3x3, positive definite and symmetric. I need to find maximum x(1) over the ellipsoid. Can I maximize x(1)^2 by taking partial derivative of the ...
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A question on relating $N$-Sphere with a $(N-1)$-cell in $\mathbb{R}^{N-1}$

Let there be a $N$-Sphere in $\mathbb{R}^N$. Every point in it is a unit vector in $\mathbb{R}^N$. Every real valued function $f$ defined on this sphere accepts a unit vector $\hat{a}\in\mathbb{R}^N$ ...
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32 views

What 's conditions on open set related to connected neighborhood of boundary

I have a question: Suppose $D$ is an open set in $\mathbb{R^n}$ and topological boundary $bD$ is an embedded submanifold of $\mathbb{R^n}$. For each $p\in bD$, we want to have an open neighborhood ...
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Invariant in geodesic

What in general is invariant in geodesic in terms of parameters $u$ and $v$ ( or functions on which they depend) and their derivatives in integrated form? For a surface of revolution, Clairaut's ...
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General expression of smooth sections of tensor bundles.

On page 317 of John Lee's Smooth Manifolds it's said that if $(x^i)$ are local coordinates on a smooth manifold $M$, then sections of the tensor bundle $T^kT^*M=\bigsqcup_{p\in M}T^k(T^*_pM)$ over a ...
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What is the notation for pull-back and push-forward of an exponential map?

So there is a nice notation for a one-parameter group of transformations $\Phi_t$ corresponding to its infinitesimal generator $\boldsymbol X$: $$\Phi_t = \exp \left(t \boldsymbol X \right)$$ But ...
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Arc length for a function $f:\mathbb{R}^2 \to \mathbb{R}^2.$

Assume $f:\mathbb{R}^2 \to \mathbb{R}^2$ is $C^1.$ Is there a formula for the length of the subset of $\mathbb{R}^2$ given by $$ \{f(x,y) \in \mathbb{R}^2:a_1\leq x\leq b_1, a_2\leq y \leq b_2\} ? $$ ...
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Topology of biological compartments

In the field of cell biology, there is a general sub-field concerned with the topology of organelle membranes, and a key focus remains on how these dynamic membranes deform and interact with cellular ...
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49 views

Parallel Transport and Christoffel Symbol

For two near by points in General Theory of Relativity. The change in the vector components when parallel transported is given by Now, since the parallel transport change must depend on the path ...
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Laplacian in arbitrary spherical coordinates?

assume that $f: \mathbb{R}^n \rightarrow \mathbb{R}$ is a function that just depends on the radius vector $r$, so no angular dependence(!). Can we say what $\Delta f$ is in arbitrary coordinates $r ...
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Pushforward of a volume form

Let $X$ be a complex projective manifold with semi-ample line bundle $ K_X$ . Assume that $f: X\to X_{can}\subset \mathbb CP^N$ , and $f^{-1}(s)$ is nonsingular fibre, then I am looking for a proof ...