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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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 ...
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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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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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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
39 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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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 ...
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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 ...
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1answer
83 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 ...
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37 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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35 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
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1answer
43 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
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1answer
52 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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34 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 ...
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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 ...
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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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21 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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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 ...
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1answer
26 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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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 $\gamma(\mathbb{R})$?
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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
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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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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1answer
45 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. ...
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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 ...
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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 ...
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1answer
26 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 ...
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2answers
115 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
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0answers
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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 ...
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2answers
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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
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1answer
37 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
36 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 ...
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1answer
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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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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
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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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1answer
23 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 ...
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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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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?
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Are stable manifold for gradient flows embedded submanifold?

Generally, the stable manifolds $W^s(p)$ of a diffeomorphism $\phi:M\to M$ is no embedded submanifold. The injective immersion $$ E^s:T_p^sM\to M $$ does not need to be a homeomorphism onto its image ...
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1answer
80 views

Differential in Lie groups

I am trying to make sense of the Lie group machinery and relate it to the calculus. Suppose that $\psi(t)=\phi(s)\phi(t), s, t \in I$. Where $\phi(t)$ is a one-parameter subgroup of the Lie group ...
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The variational bicomplex with dependent fields

I would like to understand a certain approach to variational problems that I've seen in the physics literature. In particular, I'd like to express it in terms of the variational bicomplex. However, ...
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66 views

Why Shape space is manifold?

In Shape analysis, often shape is considered as continuous parametrized closed curve and the shape space as Hilbert Riemannian manifold. Can any one help me to understand, why the shape space is ...
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51 views
+100

Does Stokes' Theorem hold on spaces with singular points?

I have come across the question whether Stokes' theorem holds also on orbifolds. Let us take the simple case of $T^2/Z_2$ with a one-form $A$, then the question becomes: For a region $\Gamma$ with ...
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1answer
56 views

Manifold allowing function with two critical points is sphere

The only closed manifolds which allow a function with two (maybe degenerate) critical points are spheres. In dimension 2 it is quite easy to prove, but what is about higher dimensions?
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1answer
65 views

How can I visualize principal bundles?

I'm trying to learn how to think about principal bundles where the fibre is a lie group with local trivialization $ϕ^{-1}_i:π(U_i)→U_i×G$ . For example $ϕ^{-1}_i:π(S^2)→S^2×U(1)$ (if that makes sense) ...
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2answers
43 views

Holomorphic Sphere $ S^2$ with (-1) self intersection number

What is the meaning of Holomorphic Sphere $ S^2$ with (-1)- self intersection number in intersection theory. Can we draw such sphere?
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1answer
49 views

Reparametrize by arc length and find the signed curvature

Let $$g(t)=(e^t \cos(t),e^t \sin(t))$$ Reparametrize $g(t)$ by arc length starting at $0$, then find the signed curvature of the unit-speed reparametization. I found that the ...