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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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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1answer
27 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 ...
4
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2answers
166 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 ...
2
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1answer
34 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 $f$ ...
5
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0answers
56 views

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 ...
4
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1answer
49 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 ...
2
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0answers
19 views

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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0answers
18 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 $F:B_2\backslash \{0\}\to B_2\backslash B_1$ ...
2
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2answers
54 views

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 ...
2
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0answers
26 views

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
24 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 ...
2
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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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0answers
36 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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0answers
36 views

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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1answer
40 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 ...
3
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0answers
37 views

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 ...
1
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1answer
24 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 ...
6
votes
1answer
84 views

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 ...
1
vote
1answer
40 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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0answers
36 views

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 ...
3
votes
1answer
44 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 ...
2
votes
1answer
54 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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0answers
36 views

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 ...
0
votes
1answer
43 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 ...
0
votes
0answers
25 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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0answers
23 views

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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0answers
13 views

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 ...
0
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1answer
27 views

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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14 views

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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0answers
31 views

Immersion, but no embedding [closed]

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 $\gamma(\mathbb{R})$?
0
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0answers
29 views

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 ...
1
vote
1answer
29 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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0answers
23 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 ...
1
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1answer
47 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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0answers
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. ...
0
votes
1answer
36 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 ...
0
votes
0answers
18 views

Verifying hypothesis (definition of Normal vector of a curve)

Let $f:I\subseteq \mathbb R\to \mathbb R^n$ a vector valued function. When we define the normal unit vector as: $N=T´(t)/||T´(t)||$, $T´(t)\neq 0$ $\forall t\in I$ ($T(t)$ is the unit tangent vector ...
1
vote
1answer
27 views

Geodesic completeness of a Lie group

Let $G$ be a Lie group and $\rho$ some left(right, bi)-invariant Riemannian metric on $G$. Is it possible to say for which $\rho$ an underlying manifold $G$ is geodesically complete (maybe for every ...
1
vote
2answers
117 views

Curvature of curve not parametrized by arclength

If I have a curve that is not parametrized by arclength, is the curvature still $||\gamma''(t)||$? I am not so sure about this, cause then we don't know that $\gamma'' \perp \gamma'$ holds, so the ...
2
votes
0answers
26 views

Geodesics on a perturbed submanifold of $\mathbb{R}^m$

Let us consider $M$, a Riemannian manifold of dimension $n$, isometrically embedded in $R^m$. Let us consider a geodesic $\gamma$ on $M$. Now, let us "perturb" (in other words, change slightly the ...
4
votes
2answers
72 views

The category of vector fields on smooth manifolds

In my differential geometry lecture today we learnt about the push-forward of a vector field by a diffeomorphism. I know some basic category theory and I noticed a functor popping up. Here's what I've ...
3
votes
1answer
39 views

Differentiability of chart functions

In the definition of a smooth (or $C^k$) manifold the charts $\varphi: U \to \mathbb R^n$ are assumed to have the property that for any two of them $\varphi \circ \psi^{-1}$ is smooth ($C^k$). Does ...
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1answer
37 views

Is every smooth function Lipschitz continuous?

Is every function of class $C^∞$ also (locally) Lipschitz continuous? If so, how can this be proven?
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1answer
24 views

Can a bilinear map on smooth functions induce a bilinear map on tensor fields in this way?

Given a manifold $M$ with a bilinear bracket $(\cdot,\cdot) : C^\infty(M) \times C^\infty(M) \to C^\infty(M)$ can it induce a bilinear map for pairs of tensor fields of different ranks in a way ...
0
votes
1answer
24 views

cup product of stiefel-whitney class

Let $\xi$ be a vector bundle. Let $w(\xi)$ be the total Stiefel-whitney class. Let $\bar w$ be the dual Stiefel-whitney class. In John Milnor's Characteristic class book, page 40-41 Chap.4, ...
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votes
0answers
26 views

Produce one smooth curve on one triangle mesh

I hope to get one smooth curve on one triangle mesh. I get one path on the mesh at first. The path consists of vertices of the mesh. I can see the path from the image below. Each one green dot ...
0
votes
1answer
35 views

Prove that there are no complete regular minimal surfaces lying above a paraboloid

Prove that there are no complete regular minimal surfaces lying above a paraboloid contained in $U=\{(x,y,z) \in \mathbb{R}^3 : a(x^2+y^2)<z\}$. Here $a>0$. I've had this problem on my mind ...
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vote
1answer
25 views

How to find tangent cone in singular point?

How to find tangent cone in singular point of surface? For example, considering surface in $\mathbb{R}^3$ given by equation $x^2z=y^2$, what is it's tangent cone in the origin? UPD:By tangent cone ...
1
vote
1answer
42 views

Conformal transformation of metric on $\mathbb{R}^n$

Let us define the following metric on $\mathbb{R}^n$: $$ g|_v(X, Y) := e^{-|v|^2} \langle X, Y\rangle,$$ where the brackets denote the standard scalar product. How does the resulting manifold look ...
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0answers
29 views

semi-positiveness of canonical line bundle under the condition Kodaira dimension be positive.

Let $M$ be a projective variety with positive Kodaira dimension, then why the canonical line bundle is semi-positive?. Is there any reference?