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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Unit Disk Regular Surface?

I am having trouble proving these two problems: 1) is $\{(x,y,z)\in \mathbb{R}|z=0, x^2+y^2\leq1\}$ a regular surface? I say no because the closed unit disk is a closed surface, so we cannot ...
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Exterior derivative of local basis element $dx^k$ is zero

Let $M$ be a smooth manifold and let $\omega = \sum_{(i_1, \dots, i_n)}f_{(i_1, \dots, i_p)} dx^{i_1} \wedge \dots \wedge dx^{i_p}$ be a differential $p$-form. Let $d$ denote the exterior derivative. ...
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Tangent space of a manifold is a vector field?

This is a follow up question on an answer to my previous question. Let $M$ be a smooth $n$ manifold and let $U\subseteq M$ be a domain. Let $T_xU$ denote the tangent space to $U$ at point $x$. Let ...
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Question regarding character varieties on a torus with compact gauge group

Let $G$ be a compact, connected Lie group. Let $x, y \in G$ be an arbitrary pair of commuting elements. Is there necessarily a torus $T \leq G$ containing $x$ and $y$? Apparently not: Commutativity ...
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Torsion vs Curvature of a curve

I was wondering if there is a simple explanation of the torsion and curvature in $\mathbb{R}^3$ of a curve. The curvature measure somehow the acceleration perpendicular to the tangent vector, but ...
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Two surfaces are not isometries of each other, but have the same Gaussian Curvature

How can you show that two surfaces are not isometries of each other, but have the same Gaussian Curvature. For example, I see that: the helicoid given by X = (ucosv, usinv, v) & the ...
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How to Compute the Torsion and Curvature of a Parametric Curve

So I have a parametric curve $\bf{r}=${$x(n),y(n),z(n)$} such that the functions $x(n)$, $y(n)$ and $z(n)$ are polynomials of $4$-th degree. I have several of these curves, and I want to calculate the ...
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Finding global sections of a vector bundle over a compact manifold which generate each fiber

The following is an exercise I am doing for review for a midterm exam in differential geometry. Question: Let $\pi:V\to X$ be any real differentiable vector bundle of rank $r$. Assume that $X$ is ...
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Bianchi identity number of independent equations

What is the number of independent equations of the second Bianchi identity: $$R_{abcd;e}+R_{abec;d}+R_{abde;c}=0$$
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Statistical Inference, Differential Geometry and Entropy

Context: Statistical Inference and Differential Geometry Let's consider a generic $ p(x;\theta) $ distribution with $ \theta $ Parameters Vector, it is obvious that $$ \int p(x; \theta) dx = 1 $$ ...
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Application of constant rank

Let $M^m$ and $N^n$ be differentiables manifolds, where $m$ is dimension of M and $n$ is dimension of $N$. If $f:M^n \to N^n$ is smooth map, with constant rank, show that: a)If $f$ is injective ...
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Find the area of parallel surface

Q: Consider a surface $M$ with regular parametrization $X:U_{open}\subset \mathbb{R}^2 \rightarrow \mathbb{R}^3$ and define the parallel surface $M_t$ by $$Y(u,v)=X(u,v) + tN(u,v)$$ where $N(u,v)$ ...
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If $\operatorname{Def} X = \frac{1}{2} \mathcal L_X g$ then what is $\operatorname{Def}^* \operatorname{Def} X$?

Let us define a deformation operator $\operatorname{Def}$ on a Riemannian manifold $(M,g)$, acting on divergence-free vector fields as $$ \operatorname{Def} X = \frac{1}{2} \mathcal L_X g, $$ where ...
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53 views

What is the definition of $dx$

I have just started to study differential forms. I don't yet fully understand the definition of what a differential form is (it's a $p$-times covariant tensor field) but I know that if $U$ is an open ...
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24 views

Basic diff.geometry question: Understanding coordinate charts by example

I recently learned the notion of coordinate chart: If $M$ is a manifold and $U\subseteq M$ is an open set in $M$ then a coordinate chart would be a smooth homeomorphism $\varphi : U \to V \subseteq ...
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158 views

Can a region always be parametrized by a single function?

Can some connected region in $\Bbb R^n$, possibly with some other nice conditions on the region, always be parametrized by a single function $\vec r(x_1, x_2, \dots, x_k)$ (even if it may be easier to ...
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31 views

A question related to the topology of the level sets of a particular type of smooth functions $f:\mathbb{R}^2\to \mathbb{R}$.

Let $f:\mathbb{R}^2\to\mathbb{R}$ be a smooth function without critical points; i.e. such that $\nabla f(x)\neq (0,0)$, for all $x\in\mathbb{R}^2$. Is it true or false that all the level curves of $f$ ...
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Basic question: Condition for a map associated to a linear series to be an immersion

I am reading this set of lectures of a class by Prof. Harris. There is a theorem. Let $X$ be a Riemann surface and $\phi:X\rightarrow\mathbb{P^r}$ be the map defined by a linear series without ...
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Minimum surface between two non coaxial rings

I'm currently dealing with minimum surfaces, especially minimum surfaces between rings. I have already studied the catenoid which is the minimum surface between two coaxial rings. Unfortunately I ...
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38 views

Riemann curvature product metric

Suppose that $M=M_1 \times M_2,$ with the product metric $g= g_1 \oplus g_2.$ Let $p\in M$ and suppose that $X \in T_pM_1$ and $Y\in T_pM_2.$ I want to show that $R(X,Y,Y,X)=0,$ at the point $p.$ I ...
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Calculation in From Seiberg Witten to pseudo-holomorphic curve

I am reading the Taubes's paper: From From Seiberg Witten to pseudo-holomorphic curve. I don't know how to get the result (2.17) \begin{eqnarray*} ...
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15 views

Pullback of Euclidean metric under spherical coordinates

I came across the following problem when I was reading a paper. Let $\Omega=B_2\subset\mathbb{R}^3$, which is the ball of radius 2 centered at origin. Map $F:B_2\backslash \{0\}\to B_2\backslash B_1$ ...
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Topologies in a Riemannian Manifold

I'm studying Differential Manifolds using Manfredo do Carmo's Book (Riemannian Geometry) and although I see no mention of this in Do Carmo's book, it's really easy to see a Riemannian Manifold as a ...
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Compare three tangent vectors constructed via parallel transport, exponential map and Jacobi vector field respectively

Given a Riemannian manifold $X$, a point $x\in X$ and $u,v\in T_xX$, I wanted to compare the following three vectors in $T_{\exp_x(v)}X$. $u_1=$ The parallel transport of $u$ along the geodesic ...
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1answer
21 views

Quaternions $\Leftrightarrow $ Rotations -Conceptual question

I have an angular velocity vector analogue defined as(called Darbaoux vector some times in curve) $ \omega(s)=\frac{\mathrm{d}\theta(s) }{\mathrm{d} s}=\delta\theta(s)_x \vec{i}+\delta\theta(s)_y ...
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1answer
28 views

Getting Ricci Curvature From $g_{ab,cd}$

How does one see that $$c g_{ab,cd}\left(\eta^{ac}\eta^{bd} - \eta^{ab}\eta^{cd}\right)$$ is equal to $$(c/2)\eta^{bc}\eta^{ae}\partial_{a}\left(g_{be,c} + g_{ce,b} - g_{bc,e}\right) - ...
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26 views

Show that in these coordinates M is locally the graph $z=f(x,y) = \frac 12(k_1x^2 + k_2y^2) + e(x,y)$

Let us say that P is the origin and TpM is the tangent plane that is the xy-plane. We will let the x,y axes be the principal directions at P. Also, we will let the limit $$\lim_{(x,y)\to ...
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on special Kähler manifolds

Take Lie group $G$ with some hypotheses (compact, connected, maybe semisimple); call its Cartan subalgebra $\mathbf t \subset \mathbf g$ and Weyl group $W$. Why does the construction $\mathcal ...
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29 views

Gaussian curvature of one sheet hyperboloid

Q: Consider an one sheet hyperboloid $S$ sitting in $\mathbb{R}^3$ which defined by $x^2+y^2-z^2 =1$. Show that there is a straight line in $S$ through every point of $S$. Also, deduce without any ...
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Are the orbits of a connected Lie group acting on a vector space always embedded manifolds?

Setting: We have a connected Lie group $G$ and a smooth map $G \to GL(V)$, where $V$ is a finite-dimensional vector space. Are the orbits of $G$ on $V$ embedded submanifolds? More precisely, if one ...
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1answer
22 views

Deriving of the Jacobi bracket and the chain rule

This is from a passage that derives the Jacobi bracket from first principles. I cannot understand how the first equality works. It seems to use the chain rule and I agree with the second term but ...
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If $f\in A_k(V)$ and $g\in A_l(V)$ show $i_v(f\wedge g)=i_vf\wedge g+(-1)^kf\wedge i_vg$ - I've got the gist, not sure how to write

If $f\in A_k(V)$ and $g\in A_l(V)$ show $i_v(f\wedge g)=i_vf\wedge g+(-1)^kf\wedge i_vg$ With $A_k(V)$ being the vector space of alternating k-tensors. for $f\in A_k(V)$ for some $v\in V$ we define ...
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36 views

A simple metric question

In their article Killing vector fields of standard static spacetimes, Dobarro and Unal derived the following simple identity. Note that if $h:I→R$ is smooth and $Y,Z∈ {\frak{X}}\left(I\right)$, ...
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Rotating Frames on Curve

We are describing a curve with a moving frame. Figure 1 says about the definition of curve Figure 1 Specifications and Proprieties of the curve We can make a rotation 3D Matrix $R(s)_{3 ...
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1answer
38 views

Canonical isomorphism between vector bundle and dual?

So, we've been asked to show, given a real vector bundle equipped with a metric, that there is a canonical isomorphism from the vector bundle and its dual. Now, there's a theorem that says two vector ...
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1answer
44 views

Is every umbilic connected surface with 0 curvature cointained in a plane?

Is every umbilic connected surface $S$ with $0$ curvature cointained in a plane? I know that the answer is "yes" if we also suppose that the surface is orientable. The argument is sketched below: ...
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The uniqueness of the Einstein metric on $\mathbb CP^n$

Is the Fubini-Study metric the unique Einstein metric (up to scaling by a constant) on $\mathbb CP^n$? More restrictively, Is the Fubini-Study metric the unique Kahler-Einstein metric (up to scaling ...
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1answer
40 views

Manifold projection to 2m+1 dimensional subspace is a manifold.

Let $M \subseteq \mathbb{R}^n$ be a m-dimensional manifold. Suppose $n>2m+1$. Show that there is a projection from $M$ to a (2m+1)-dimensional subspace of $\mathbb{R}^n$ so that the image is ...
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23 views

Help with terminology

I need some help unraveling the terms that appear in the following passage. I found it in a book on some conference proceedings related to Differential Geometry. Let $f:X \to R^3$ be a smooth curve ...
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Why is the space of symmetric positive definite matrices under the affine invariante metric a symmetric space?

I consider here the space of symmetric positive definite matrices SPD(n) with the metric invariant under: $(G,M) \rightarrow GMG^t $, where $M\in SPD(n)$ and $G\in Gl(n)$. The isotrpie group of $I$ is ...
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Topology of the intersection of toric arrangement

Hope someone will help me in the solution of the following question. I'm working on some topological problem involving the topology of the intersection of some characters of the torus. I want ot find ...
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measuring curvature

Suppose you are transported to an 2 dimensional hyperbolic world, ( a plane (2 dinensional) manifold with a constant negative curvature ) the only geometrical tools you have are a ruler, a pencil, ...
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whitney class of fiber bundles over classifying space of permutation groups

Let $\Sigma_k$ be the permutation group of order $k$. Let $\rho: \Sigma_k\to GL(k)$ be regular representation by permuting basis. Let $\rho': B\Sigma_k\to BGL(k)=G_k(\mathbb{R}^\infty)$ be the induced ...
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Immersion, but no embedding [on hold]

Show that the map $$\gamma:\mathbb{R} \to \mathbb{R}², \quad\gamma(t)=(2\cos(\pi/2+2\arctan t), \sin(\pi+4\arctan t))$$ is an homeomorphism over $\gamma(\mathbb{R})$?
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If $[\omega]\in H^2(M,\mathbb Z)$ then $M$ is projective?

Let $(M,\omega)$ be a Kähler manifold with $[\omega]\in H^2(M,\mathbb Q)$ then why $M$ must be projective variety. As I know if $[\omega]\in H^2(M,\mathbb Z)$ then $M$ is projective by Kodaira theorem ...
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1answer
26 views

Equivalent definitions of partition of unity?

On Wikipedia a partition of unity is a collection of continuous maps $\varphi_i$ from a topological space $X$ into $\mathbb R$ such that for all $x$ (i) $\sum_i \varphi_i (x) = 1$ (ii) there is a ...
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22 views

Books or notes focusing more on the intersection between manifolds and topology?

I try to prepare for the qualify exams, and find that the problems of geometry part are quite interesting. In the past, I just learned some elementary things on manifolds and algebraic topology ...
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41 views

Differential-geometry textbook with solved problems

I'm looking for a textbook in differential geometry which inside has exercises with (at least) final answers. Since it's my first course in differential geometry it doesn't have to cover material (we ...
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48 views

Are the principal curvatures on a surface always smooth?

It's easy to show that the principal curvatures on a surface are smooth away from umbilic points since we may write a expression for them using the Gauss curvature and the mean curvatures, locally. ...
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1answer
34 views

Let $\pi: E \to M$ a vector bundle. Is $E$ a direct summnad of $M\times\mathbb{R}^{d}$, for some $d$?

Let $\pi: E \to M$ a vector bundle over a smooth manifold $M$. $E$ is direct summand of $M\times\mathbb{R}^{d}$, if there exist a vector bundle morphisms $f:E\to M\times\mathbb{R}^{d}$ and ...