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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Complex projective manifolds and holomorphic mappings

Let $X\subset\mathbb{P}^2$ be a complex manifold defined by a homogeneous polynomial of degree $d>3$. Let $$\phi:\mathbb{P}^1\rightarrow X$$ be a holomorphic map. Show that $\phi$ is constant. ...
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Alexandrov embedness and branch points

Let assume that $\Sigma_n$ is a sequence of compact surfaces in $\mathbb{R}^3$ of fixed genus. We assume that the surfaces are Alexandrov embedded, that is to say there exits an immersion $i_n$ from ...
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30 views

Simple question on symmetric tensors

This question seems to be silly, but i am really confused. Suppose we have a symmetric $2$-tensor $\omega$, I want to prove $$\omega(X,Y)=0\ \ \ \forall \ \ \ X,Y \iff \omega(X,X)=0 \ \ \forall \ \ \ ...
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Differential of the inversion of Lie group

Let $G$ be a Lie group and $\iota \colon G\to G$ denote the inversion. If $e$ is the identity of $G$, prove that: $$\text d \iota _e = -\text {id} _{T_e G}.$$ I understand that the differential at $e$ ...
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25 views

Quaternion Solution of the Rotation Equation

I am trying to make a connection between a 3-d vector ODE with a quaternion ODE and a possible solution in quaternion. In the following, a vector $v$ in $R^3$ is interpreted as the vector part of the ...
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66 views

Geodesic deviation on a unit sphere

No response to this on Physics Stack Exchange, so I'm hoping for better luck here. My question is, can anyone tell me where I'm going wrong trying to use the equation of geodesic ...
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39 views

Relation between a function on $N$ sphere and a function on $(N-1)$-cell.

Let $S^N$ be a unit $N$-sphere. Let $f:S^N\to\mathbb{R}$ be a function. Let $\bf{\Sigma}$ be a unit $(N-1)$-cell, consider the function $g:S^N\to\bf{\Sigma}$ such that, for any $\hat{a}\in S^N$, ...
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32 views

Non-vanishing differential forms and flatness

Let $M$ be a differentiable manifold of dimension $n$. If the tangent bundle is trivial, then the cotangent bundle is trivial, and so are its exterior powers. In other words, on a parallelizable ...
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180 views

gaussian and mean curvatures

I am trying to review, and learn about how to compute and gaussian and mean curvature. Given $\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$, how can I compute the gaussian and mean ...
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57 views

Sine-Gordon Equation application

Is it true that Sine-Gordon is satisfied for geodesics on the central Pseudosphere ( rotated surface of Tractrix)? If so, please cite text-book or article references. $$ \alpha''(s)=sin ( \alpha) ...
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Examples of qualities intrinsic vs extrinsic to a surface besides Gaussian Curvature

Gauss's Theorema Egregium states that Gaussian curvature is intrinsic to a surface, meaning that it can be "measured inside of the surface". However I can't make sense of what this really means. What ...
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65 views

Normal curvature of a circle in a plane

I have the circle $\gamma(t) = (\cos t, \sin t, 0)$ in the plane $z=0$. Now I understand that normal curvature is related to the second fundamental form, and an expression for it is ...
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39 views

Problem with a Do Carmo problem

I'm trying to solve some Do Carmo problems from his book Differential geometry of curves and surfaces. In section 1-3 prob.5.c., we have the curve: $\alpha:(-1,\infty)\rightarrow\Bbb R^2$ given by: ...
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109 views

is tangent bundle of $S^n$ an algebraic variety?

I have found somewhere that $T(S^n)$ is an algebraic variety in $\mathbb{C}^{n+1}$. But now I can not recall the explicit form of this variety and the source of this information. It will be helpful if ...
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cohomology ring of grassmannian

Let $G_k(\mathbb{R}^n)$ be the grassmannian consisting of all $k$-subspaces in $\mathbb{R}^n$. How to compute the cohomology ring $$H^*(G_k(\mathbb{R}^n);\mathbb{Z})$$ and what is the result?
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42 views

Metric tensor and einstein's notations .

Here , $\Omega \subset \mathbb R^n$. Can someone explain it to me what $F$ is ? I also don't understand how we can get the unit outer normal in the second para . Please kindly help me to understand ...
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How can I compute the area of a geodesic triangle?

How can I compute the area of a geodesic triangle in a Riemannian 2-manifold? If the Gauss curvature $K$ is constant and positive I can take the Gauss-Bonnet theorem to obtain ...
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30 views

definition of critical point defined in terms of differential map

I am having a problem understanding the definition of a critical point in do carmo's Differential Geometry of Curves and Surfaces. He notes in page 58 that a point $p \in U $ is a critical point of ...
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59 views

Is an configuration space contractible?

Let $\mathbb{R}^\infty$ be the union $\cup \mathbb{R}^n$(with inclusions). Let $F(\mathbb{R}^\infty,k)$ denote the $k$-th configuration space of $\mathbb{R}^\infty$, i.e., the subspace of ...
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21 views

Identity concerning push forward of two vector fields

How would you prove the identity $\displaystyle \frac{\partial}{\partial s}\Psi_{s^*} \mathbb{X} = (-1)L_{\mathbb{Y}}\Psi_{s^*}\mathbb{X}$ where $\Psi_{s}$ is the flow of $\mathbb{Y}$ and ...
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16 views

about moments of a uniform distribution on a high-dimensional ball

I need to understand how the following integrals depend on the dimension $d$; the result should be about a (negative) power of $d$. Let $\mathbb{B}^d$ be the $d$-dimensional ball of radius $1$, ...
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37 views

tangent bundle and normal bundle

I have a problem about tangent bundle. It is known that the tangent bundle of most manifolds is not trivial: for example, the tangent bundle for $S^2$ is not $S^2\times \mathbb{R}^2$. However, for a ...
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Characterization of the exterior derivative $d$

In the paper Natural Operations on Differential Forms, the author R. Palais shows that the exterior derivative $d$ is characterized as the unique "natural" linear map from $\Phi^p$ to $\Phi^{p+1}$ ...
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defining smooth functions on manifolds *without* smooth chart transitions

Let $M$ be a topological manifold, covered by an atlas of charts ${(U,\phi_U)}$ (which are homeomorphisms into Euclidean space), and let $p\in M$. Say a function $f:M\to\mathbb{R}$ is smooth at $p$ if ...
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How do I map Df(w) to it's [lie] group/algebra representation?

E.G. For $p,w\in(\mathbb{R}^3,+,\times_\vartheta)$ with $(\mathbb{R}^3,+)$ a vector space and with $p=(r,s,t)$, $w=(x,y,z)$ where we have $p\times_\vartheta ...
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17 views

Maximize first co-ordinate on general ellipsoid

I have an ellipsoid of the form x^TAx=k , where A is 3x3, positive definite and symmetric. I need to find maximum x(1) over the ellipsoid. Can I maximize x(1)^2 by taking partial derivative of the ...
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63 views

show $\omega$ is exact form

Let X be the region $\mathbb{R^3}-(0,0,0)$ and f(x,y,z) is $C^\infty$ function on X. Also $\omega$ is 1-form $f(x,y,z)(xdx+ydy+zdz)$. if $f$ can be expressed in the form $f(x,y,z)=h(r)$ when ...
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27 views

Sign of Laplacian in a N-Dimensional Space Under Other Specific Conditions

Consider that $f: \mathbb{R}^N \to \mathbb{R} $ is infinite times differentiable and is smooth over the entire domain. Take $N$ to be large. Also at point $\mathbf{x_0}$: $$ ...
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22 views

Finding an isometry that maps one circle to another.

I have a problem goes as follows: Consider the unit speed curve $$\boldsymbol{r}(s)=\left(\frac{4}{5}\cos(s),1-\sin(s),-\frac{3}{5}\cos(s)\right).$$ Find an isometry $f$ such that ...
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A Problem from Docarmo's Differential Geometry

The following is a (may be simple) problem from Docarmo's Differential Geometry. Let $\alpha\colon (a,b)\rightarrow \mathbb{R}^3$ be a parametrized curve which do not pass through origin. If ...
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43 views

Use contraction mapping theorem to prove

Help! I am taking a math course, and I just can't figure out this proof: Let $\alpha,\beta\in R^n$, $a\in R$, and $A$ be an $n\times n$ nonsingular matrix. Use contraction mapping theorem to prove ...
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Hessian is proportional to the metric everywhere

Let $(\Omega^{n+1},g)$ be a compact Riemannian manifold with smooth boundary. Let $f\in C^{\infty}(\bar{\Omega})$ satisfies $\operatorname{Hess}f=\frac{1}{n+1}g.$ Suppose the minimum of $f$ occures at ...
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Differential Geometry Intuition Question

Apologies if I get the notation wrong. Still learning this stuff. Suppose I have a 2 dimensional Riemannian manifold $\mathcal{M}$ that is covered by a single chart: $\phi: \mathcal{M} \rightarrow ...
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Exponential of a power of the differential operator

In relation to this question: Exponential of a polynomial of the differential operator Is there an expression for $\exp(aD^n)f(x)$ similar to $\exp(aD)f(x)=f(x+a)$?
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What does this operator $\odot$ mean

I read this about the second fundamental form in Wikipedia and I’ve no idea what does $\odot$ mean? Does anybody know? $$II=-dN\cdot dP=\omega^3_1\odot\omega^1+\omega^3_2\odot\omega^2$$
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defining smooth functions on smooth manifolds

The standard approach to defining smooth functions $f:M\to\mathbb{R}$ on a topological manifold $M$ equipped with a smooth structure (i.e., a maximal smooth atlas) $\mathcal{A}$ is the following. Say ...
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Problem about Ricci Flow

On page 12 of "Lectures On Ricci Flow" by Peter Topping is written: In two dimensions, we know that the Ricci curvature can be written in terms of the Gauss curvature $K$ as $Ric(g) = Kg$. Working ...
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A question on relating $N$-Sphere with a $(N-1)$-cell in $\mathbb{R}^{N-1}$

Let there be a $N$-Sphere in $\mathbb{R}^N$. Every point in it is a unit vector in $\mathbb{R}^N$. Every real valued function $f$ defined on this sphere accepts a unit vector $\hat{a}\in\mathbb{R}^N$ ...
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map between classifying spaces induced by group homomorphism

Let $\Sigma_n$ be the permutation group of order $n$. Then the regular representation of $\Sigma_n$ gives an injective homomorphism $f:\Sigma_n\to O(n)$. Why $f$ induces a map between their ...
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Calabi-Yau manifolds and immersion in real space [closed]

I'm reading some papers how to test extra dimensions in LHC experiments and they suggests CY manifolds as starting point. Is it possible that accelerator itself is made in higher-dimensional geometry ...
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46 views

Visualizing covariant derivative

I am trying to show an intuition behind covariant derivatives on a sphere, but I realize my intuition is probably incorrect. I first want to show why the covariant derivative of the tangent vector ...
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1answer
35 views

Minkowski space is locally Euclidean?

The Minkowski spacetime $\mathbb{R}^{1,3}$ is said to be a manifold (isomorphic to $SO^{1,3}$. But according to the definition of a manifold it should be locally euclidean. However, this seems to be ...
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Invariants of shape operators?

Let $S:V\longrightarrow V$ be a Linear Transformation, then the Characteristic polynomial of $S$ and therefore its Coefficient are invariant. Except the first and the last Coefficient that we know ...
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local diffeomorphism

I hope this finds you all well. Could someone give me an example of a local diffeomorphism from $\mathbb{R}^p$ to $\mathbb{R}^p$ (function of class say $C^k$ with an invertible differential map in ...
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Why is the Leibniz rule a sufficient ingredient in the construction of the tangent space?

This is a very soft question, but I am wondering if anyone can shed light on why it is that the product rule (and linearity) provide exactly the right requirement for the space of derivations to be ...
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Yarn-like functions

When wrapping yarn around a ball you cannot make sharp turns or the yarn will fall off. If we think of the yarn as a curve on the surface of the sphere, we would say it must have curvature less than ...
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3answers
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Books on differential geometry in the cases $n=2$ and $n=3$

I'm interested in learning the differential geometry of standard, "physical" space, that is $\mathbb R^2$ and $\mathbb R^3$. The sort of problems that were studied in the 18th and 19th century... ...
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31 views

What 's conditions on open set related to connected neighborhood of boundary

I have a question: Suppose $D$ is an open set in $\mathbb{R^n}$ and topological boundary $bD$ is an embedded submanifold of $\mathbb{R^n}$. For each $p\in bD$, we want to have an open neighborhood ...
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700 views

Want to learn differential geometry and want the sheaf perspective

I would like to learn some differential geometry: basically manifolds, differentiable manifolds, smooth manifolds, De Rham cohomology and everything else that is pretty much part of a course in ...
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93 views

The relation between geodesics and distances on a Riemannian manifold

My question is about computing the distance between two points in a Riemannian manifold. Suppose that $(M,g)$ is compact so that it is geodesically complete and geodesically convex. Let ...