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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Curvature and Torsion explained

Let $\alpha: I \to \Bbb R^3$ be a curve parameterized by arc length $s$, with curvature $k(s)\ne 0$, for all $s\in I$ $$\alpha(s) = \left(a\cos \frac sc, a \sin \frac sc, b\frac sc\right),\quad s\in ...
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Differential Geometry concept verification

If I have a regular parameterized curve: $$\alpha(t)$$ The curvature, $k(t)$, is precisely $\|\alpha''(t)\|$, The normal vector, $n(t)$ is found by looking at $\alpha''(t) = n(t)k(t)$ The binormal ...
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Singularity in Ricci flow vs Ricci soliton

In the paper "The formation of singularity in Ricci flow" Hamilton studied systematically the possible singularities of the flow.My question is why it is important to classify Ricci solitons in order ...
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Level sets on $SU(n)$

Given $G \in SU(n)$, what are the level sets of the function $F(V) = |tr(G^{\dagger}V)|^2$? Can they be written only in terms of abstract linear maps, not in terms of the components of $V$ and $G$ in ...
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30 views

A detail in the proof orientable manifold admits exactly two orientations

Let $M$ be a real manifold and let $\{(U_\alpha,\phi_{\alpha})\}_{\alpha\in I}$ $\{(V_\beta,\psi_\beta)\}_{\beta\in J}$ be two oriented atlases. Let's define, for $p\in U_\alpha\cap V_\beta$, ...
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16 views

Locus of points on a rotating line ; points differently ordered

A line rotates about a fixed point $O$ with ordered points $P,O,M $, while $ M $ is moving along this line $POM$. Find locus of points $ P ,M $ if $ MP^2- OM^2 = T^2 $ constant for all inclinations ...
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18 views

Classification of parallel spinors on the torus?

The $2$-torus $T^2$ has four different spin structures, typically parametrized by $\delta \in Z_2^2$. How many tuples $(\delta, g, \varphi)$ exist such that $\delta$ gives a spin structure on $T^2$, ...
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Is the parametrization $(t^3,t^6)$, a reparametrization of $(t,t^2)$?

I have gathered that it is not. But I find it to obey all the conditions of being a reparametrization. Definition given in comments :P
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standard unit tangent vectors to the unit sphere

I am reading the following I am having trouble in understanding how one can compute standard unit tangent vectors to $S^2$, i.e. the following Could anyone explain to me how $\theta$ and $\phi$ ...
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55 views

Problems understanding this proof

This is an extract from Duistermaat's Fourier integral operators. I'm having a hard time understanding the proof. My questions are three: How do I use the implicit function theorem to ...
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35 views

Reference for theorems in Hirsch

In Hirsch's differential topology, we find the following theorems on page 31: 4.1 Theorem: Let $M$ be a $C^r$ $\partial$-manifold and $N$ a $C^r$ manifold, $r \geq 1$. Let $f : M \to N$ be a $C^r$ ...
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Showing that continuous forms are zero on a $\mathscr{C}^1$ simplex $\Psi$ , if all smooth forms are zero on $\Psi$.

Question: Is the guess below correct? EDIT: There haven't been any responses yet; I wonder if the question needs to be improved somehow... Forms and simplexes are as in Rudin Rudin Principles of ...
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14 views

Applications of normal coordinates?

I am looking for applications of normal coordinates (from the exponential map) on a Riemannian manifold, since I am trying to familiarize myself with this notion. Presently the only one that I know ...
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39 views

Mistake in book on symplectic topology?

I just read the proof of the non-squeezing theorem in "Introduction to symplectic topology" by Mc Duff and Salamon. The thing that is strange is that they say: Let $\Psi$ be the linear transform ...
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38 views

Curvature of a level set

I am using the level set method for image segmentation. In particular, the segmentation boundary $C(x, y)$ is represented as the zero level set of a level set function $\phi(x, y)$. As working on ...
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17 views

find the tangent space of hyberboloid?

How can I find the tangent space of the hyberboloid $$ x^2 +y^2 -z^2=a$$ for $$a>0$$ in the given point: $$(\sqrt{a},0,0)$$?
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27 views

Implicitization of Parametric Curves

I've got a 3D parametric, smooth, simple, and closed curve given by $\sigma(s) = (\sigma_1(s),\sigma_2(s),\sigma_3(s))$ where $\sigma_1(s)$ and $\sigma_2(s)$ are given by trigonometric functions of ...
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29 views

2-dimensional Riemann Manifold

I am looking for a proof of the theorem that states that any 2-dimensional Riemann Manifold is conformally flat in the case of a metric of signature 0, following through with Problem 6.30 in the text ...
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Computation in Wikipedia's article “Riemann Curvature Tensor”

This Wikipedia article explains how the Riemann curvature tensor is a measure of the failure for a tangent vector to parallel translate back to itself along an infinitesimally small loop. The article ...
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38 views

length of continuously differentiable curves

I saw that the length of a continuously differentiable curve $\gamma$ in $\mathbb{R}^n$ with $\gamma(t) \neq 0$ is defined as $\int_a^b |\gamma^{'}(t)|dt$, as can be found here ...
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42 views

Exponential map only for matrix Lie algebras?

Recently, I stumbled over some proofs in Lie algebra theory and noticed that they often use the notion of an exponential map $e^{t \zeta}$ for $\zeta \in \mathfrak{g}$ such that $e^{t \zeta} \in G$ ...
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42 views

Is there a deRham (co)homology for vector-valued differential forms?

Is there a deRham (co)homology for vector-valued differential forms? The deRham (co)homology of differential forms has been well-discussed and well-founded, along with the fact that for the exterior ...
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40 views

How to tell complex structures apart

Complex structures are rigid, yet weirdly flexible. For example, the Riemannian mapping theorem says that every non-empty simply connected open subset of $\mathbb{C}$ that is not $\mathbb{C}$ is ...
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Adjoint and coadjoint orbits

I just read that for the Lie algebras $\mathfrak{gl}(N),\mathfrak{sl}(N),\mathfrak{so}(N),\mathfrak{sp}(2N)$ the adjoint and coadjoint orbits coincide. Now, the adjoint orbits are $O_{\xi} = ...
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25 views

Stabilizer subgroup in adjoint action

Given $b \in \mathfrak{su}(n)$, how can I find the stabilizer $\text{stab}(b)$ for the adjoint action of $SU(n)$ on $\mathfrak{su}(n)$ given by $Ad_U(b) = UbU^{\dagger}$ without using coordinates? The ...
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Please would someone check my answer to this exercise on vector fields along maps?

I believe I solved the following exercise and would appreicate it greatly if someone could check my answer: Let $U$ be an open neighbourhood of $0 \in \mathbb R^2$ and let $f = (f_1, f_2) : U \to ...
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69 views

Differential forms- riddle me this.

I stumbled over this answer on math.stackexchange Let $x$ be a point in $\rm M$. Then because $\omega$ is non degenerate at $x$, the antisymmetric matrix $\omega_x$ has full rank on the tangent ...
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20 views

Why $exp(0_{T_eG})=e$, where $exp$ is the exponential map of a Lie group?

I wonder if this fact is true: I consider the exponential map of a Lie group $G$. $$exp: \mathfrak{g} \rightarrow G.$$ Is it true that $exp(0_{T_eG})=e$, where $e$ is the identity element of $G$? ...
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41 views

Topology of statistical manifolds

I am currently working with statistical manifolds. Roughly, a statistical manifold is a set of distribution parametrized by a set of parameters. However i have trouble finding more precise definition. ...
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14 views

Transitive group actions on Principal bundles

I have a question in regards to page 107 of Kobayashi & Nomizu's Foundations of Differential Geometry Setup: Let $\pi:P\rightarrow M$ be a principal bundle with structure group $G$ whose Lie ...
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28 views

Equivalence of two integral conditions

Consider the following parametrization of the unit ball in $\Bbb{R}^3$: \begin{align}T:(0,1)\times (0,\pi)\times (-\pi ,\pi)&\to \Bbb{R}^3 \\ r,\theta,\phi &\mapsto(r\sin \theta\sin \phi, r ...
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Is there any reason why both derivatives should be non zero?

Let $f = (f_1, f_2) : \mathbb R, 0 \to \mathbb R^2 , 0$ be smooth and such that $n = \min (ord(f_1), ord(f_2)) < \infty$ where $ord(f) = \min \{n \in \mathbb N_{>0}\mid {\partial^n \over ...
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27 views

Curve shortening flow and strong maximum principle

I am in particular uncertain about how the strong maximum principle is used in the argument below. Could someone please clarify and add more detailed explanations. Thanks So assume we have a regular ...
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44 views

Prove that an homogeneous and isotropic Riemannian manifold has constant sectional curvature

I have problems in proving that an homogeneous and isotropic Riemannian manifold has constant sectional curvature. This is my attempt: By definition, the manifold $M$ has constant sectional ...
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30 views

Line integral and differential forms

Let $df = \frac{\partial f}{\partial x_1 } dx_1 + \frac{\partial f }{\partial x_2 } dx_2$ be a $1-form.$ I know that that the line integral (along a curve $\gamma:[0,t] \rightarrow \mathbb{R}^2$) is ...
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Inverse mappings and transition functions

Could somebody tell me if this is correct? I'm trying to understand mappings and inverse mappings in introductory differential geometry. The transition functions baffle me. Suppose our manifold of ...
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32 views

Basic exercise differential forms

I have to show that the space of q-differential forms $\Omega^q(U)=\{0\}$ if and only if $q>n$ or $q<0$. Any ideas?
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Riemann Tensor in Particular Frame

I'm trying to reproduce a calculation which requires computing the Riemann tensor in a particular frame specified by some vierbein $e_a$. I have a complete expression for the spacetime metric in some ...
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23 views

Symplectic form on a Hilbert Space is Closed

Let $\mathcal{H}$ be a Hilbert space over $\mathbb{C}$. Define a new vector space, $V$, over $\mathbb{R}$, which has, on the level of sets, $V = \mathcal{H}$ and for scalar multiplication (only with ...
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25 views

Is the dimension of a smooth manifold an invariant of the underlying set in it?

Let $M$ be a smooth manifold, $S$ a set and $f:M\to S$ a bijection (assuming of course, that such a function does exist). It's an easy exercise to show that $S$ can be given a differentiable ...
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26 views

A theorem of Cartan

My question is regarding when the polarmap defiened from an isometry $L:T_pM\rightarrow T_{\bar{p}}N$ extends to an isometry of the normal neighborhhods on which it is a diffeomorphism. I have come ...
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Required minimum number of points on boundary of minimal surface

What is the minimum number of points required to uniquely determine a minimal surface in 3-Space? Four? Five? If six or more points on boundary are given it gives rise to over-determination.. right?
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28 views

Scaling of minimal surfaces

After scaling and suitable Euclidean motions every rigid minimal patch can be placed on a unit catenoid of revolution $ x^2 + y^2 = c^2 \cosh^2 (z/c), c=1.$ with full area contact. Is the statement ...
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Divergence and formal adjoint operators: are they bounded/continuous?

Let $(M,g)$ be a smooth Riemannian manifold. The divergence operator is the map \begin{align*} \delta_g:\Gamma^k(S^2M)&\rightarrow\Gamma^{k-1}(T^*M)\\ ...
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Natural derivative of Vector Fields on manifolds

I'm learning about connections and my book says that there is no natural derivative for a vector field on a manifold. Wouldn't it be possible to cook up a connection by just letting $\nabla_{v_p}X = ...
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Normal Vectors to Action of Orthogonal Group

Let $X\in\mathbb{R}^{n\times r}$ be a fixed matrix with orthogonal columns, and let $U\in\mathbb{R}^{n\times r}$ be given. Because the group of orthogonal $r\times r$ matrices, $O(r)$, is a compact ...
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decomposition of a closed surface

I know that I can decompose an hyperbolic closed surface of genus $g>1$ into $2(g-1)$ pants bounded by 3 geodesics. It seems reasonable to think the same can be done for a closed surface of genus ...
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22 views

The configuration space of a linkage

I am now reading the book 'geometry and billiards' by Tabachnikov. In this book, he has such a problem: Can anyone solve the case (c)?
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Consistent and rigorous definitions of curve, arc and path

While reading many books, textbooks or Internet and actually I'm getting very much confused about the issues related to definitions of arcs, curves, paths, their parametrizations and curve images. Is ...
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Computation of Hyperkahler Metric using kahler forms

I am trying to compute a hypekahler metric using its Kahler forms. We can expand the $\omega_{\alpha}$ as $\omega_{\alpha}={h_{\alpha}}_{ab} dx^a\wedge dx^b$ in which $x^a \in (u,\overline{u};p,q)$ ...