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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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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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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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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7 views

Singularities of composit function

Given a smooth, compact manifold $M$ (of dim 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 of ...
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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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Confused in some basic concept about Gauss Map $dN_p$

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

Examples which illustrate the local immersion and submersion theorems

I am looking to gain a better understanding of both the local immersion/submersion theorems via non trivial examples (but also nothing too complicated), which would help illustrate the parts of their ...
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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
34 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
18 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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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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16 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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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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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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Extending the unit tangent to an analytic function

Suppose $\Gamma\in\mathbb{R}^2$ is a smooth, simple closed curve and denote its unit normal vector(say outward) at each point $z\in\Gamma$ by $T(z)$. Under what assumption on the boundary curve, $T$ ...
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29 views

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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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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31 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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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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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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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 $$A = \{(x,y,z) \in \mathbb{R}^3 | x^2+y^2+z^2=1, z\geq 0\}.$$ ...
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Curvature Relationship with Norm of the Curve at the Point of Maximum Norm

Continuing on my series of questions, for those following, is the following question: Let $\alpha: I \to \mathbb{R}^3$ be a regular curve. Suppose that for some $t \in I$ the distance from the ...
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Differentiable, Parametrized Curve for Trace $y = |x|$

my professor gave practice problems for an upcoming midterm, and one is to find a differentiable parametrized curve with trace $y = |x|$ with $-1 \leq x \leq 1$. My thinking is that I should find a ...
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The principle curvature of the half cylinder $\{(x,y,z): y^2 + z^2 =1, z>0\}$

I have the half cylinder $\{(x,y,z): y^2 + z^2 =1, z>0\}$ and I want to calculate the principle curvature of this. I know the principle curvatures of this cylinder S at a point p , denoted $k_1$ ...
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Physical meaning of cofactor and adjugate matrix

I like the way there a physical meaning tied to the determinant as being related to the geometric volume. Since the determinant can be calculated through Laplace's formula where the cofactor matrix is ...
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$\mathbb{R}^3$ not diffeomorphic to $\mathbb{R}^3\setminus \{0\}$

I have to show, that $\mathbb{R}^3$ is not diffeomorphic to $\mathbb{R}^3\setminus \{0\}$. That means, I have to show that there are no two smooth maps $f\colon\mathbb{R}^3\rightarrow ...
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Angle between two curves on a surface

Let $\mathcal{M}$ be a surface and $\gamma_1, \gamma_2$ two smooth curves contained in $\mathcal{M}$ in natural parameterization s.t.: $\gamma_1(0)=\gamma_2(0) = p$ , ...
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resources for classical gauge theory

As a prospective grad student, I would like to get an entry level introduction to classical (i.e. non-quantum) gauge theory. Please direct me to resources suitable for a novice.
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Proving a compact Lie group admits a biinvariant metric [duplicate]

At the end of a lesson in Differential Geometry, my teacher said: Fatto, che non dimostriamo, non è difficile ma il tempo scarseggia, se $G$ è compatto possiamo sempre trovare una metrica ...
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1answer
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Is integral curve a embedded 1 dimensional submanifold of the given manifold?

I can easily see a proof that shows its going to be an immersed submanifold . (I am removing the case if the vector field at that point is 0). I am not able to see if it's a embedded submanifold or ...
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singular $1$ cube - Boundary of $2$ chain

This is an exercise from "Calculus on Manifolds" by Michel Spivack (first edition, p.100): If $c$ is a singular $1$-cube in $\mathbb{R}^2-\{0\}$, with $c(0)=c(1)$, show that there is an integer ...
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How to find the tangent space of a general submanifold?

Given a submanifold $(S,\phi)$ of a manifold $M$, how do we find the subspace of $T_pM$ that is equal to $T_pS$ for $p\in \phi(S)$. I know how to do it for level sets. Is there a way for general ...
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Write $\gamma(t) = (t,t^2,t^3)$ as a graph and a level set

My exercise is to write the twisted cubic as a graph and a level set. However, I am not sure what they mean by a graph and level set. Can anyone explain this please? Do they mean, for a graph, ...
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Every non-constant closed curve has positive period

I want to show that every non-constant closed curve has positive period, but i'm not really sure how to do this. A smooth curve $r(t): \mathbb{R} \to \mathbb{R}^n$ is $T$-periodic if $r(t+T)= r(t)$ ...
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Independence of parametrization in defining integral of differential form

This is an exercise from Spivak's Calculus on Manifolds. Questions asks the following : (Independence of parametrization). Let $c$ be a singular $k$ cube and $p:[0,1]^k\rightarrow [0,1]^k$ be ...
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Einstein notation

I'm confused about a specific issue that I have with the Einstein notation (for tensor fields on manifolds). I want to write the following thing: Let $X$ be a smooth manifold. Choosing local ...
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Conformal curvature line parametrization

While reading a paper I found a definition which is confusing me. Def: A conformal curvature line parametrization $(x,y) \to F(x,y)$ is called isothermic. I know what a conformal ...
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moving frame with maple

I have already ask this question on stackoverflow, but since it concerns as mathematics than computer science, I ask it here too. I would like to make a classical computation using maple. I would ...
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Proof of relation between normal of a surface and principle curvatures of surface.

If F(x,y,z) is a scalar function. Then how to prove that, $$\nabla . n = K_1 +K_2$$ where n is normal to surface of constant $F$ given as $$n=\frac{\nabla F}{|\nabla F|}$$ $K_1$ and $K_2$ are ...
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Computing Gauss's of a sphere

The vector field given as $\vec{F}=\frac{\left \langle x,y,z \right \rangle}{\sqrt{x^{2}+y^{2}+z^{2}}}$ The region $D=\left \{ a^{2}\leq x^{2}+y^{2}+z^{2}\leq b^{2} \right \}$ I've some ...
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66 views

When is a diffeomorphism analytic?

I've read somewhere "a class $C^\infty$ diffeomorphism is said to be analytic" but I forgot to write down where I read this and now I can not find it, which makes me wonder if it's true? I'm working ...
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When is the metric completion of a Riemannian manifold a manifold with boundary?

Let $(M,g)$ be a connected smooth Riemannian manifold and denote by $(M,d)$ the induced metric space following by taking topological metric to be the infimum over length of curves in the standard way. ...
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“vector” vs “point” in definition of directional derivative

Given a function $f\colon \mathbb R^n\to\mathbb R$, and given $x,v\in\mathbb R^n$, it is customary to define the "directional derivative of $f$ in the direction $v$ at the point $x$" by $$ D_v f(x) = ...
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For which Lie groups $G$ can one write $g$ as the exponential $\exp X$ of some $X \in {\frak g}$ for every element $g \in G$?

I am reading a book on matrix Lie algebras (Brian Hall's). Corollary 2.30. says that if $G$ is a connected matrix Lie group, then every element $A$ of $G$ can be written in the form ...