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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Polynomial functions on a smooth manifold

If one views $\Bbb R ^{2n}$ as the cotangent bundle of $\Bbb R ^n$, with coordinates $(q_1, \dots, q_n, p_1, \dots, p_n)$, then in order to do classical Hamiltonian mechanics on it one considers ...
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Coordinates on de Sitter space

I am trying to use a certain parametrization on de-Sitter space $dS^n$ and I am getting both the wrong scalar curvature and metric determinant. The formal definition of $dS^n$ in my work is ...
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first Chern class of pair (X,D)

Let $(X,D)$ be a pair of projective variety $X$ and $D$ is a simple normal crossing divisor on $X$ then is it correct that $$c_1(X,D)=c_1(X)+[D]$$ where $[D]$ is the current of integration
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Why do those terms vanish if the metric is Hermitian?

On this [page][1], the author says: We now define a Hermitian manifold as a complex manifold where there is a preferred class of coordinate systems in which unmixed components of metric tensor ...
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How to show the inf can be achieved by some nonnegative $u\in H^1(M)$?

When I read some about Perelman's $\mathcal W$ function, I get stuck with the red line in the picture below.Seemly, I should to read the 8.2 Existence of minimizers of Evans' PDE. But I am not sure , ...
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Basis of differential one-form confusion

I'm trying to understand one-forms in the context of general relativity. Lee (Introduction to Smooth Manifolds) says that at a point $p$ and with a vector field $X$ we define a covector field ...
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What is the difference between Differential of a map and Parallel transportation?

If $\gamma$ is a closed geodesic in $M$ with $p=c(0)=c(1)$. We have two maps, namely first is the differential of the action of the class $[c]\in \pi_1(M, p)$ on the tangent space $T_p(\tilde{M})$, ...
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Differentiating a certain vector field

Let $M^n$ be a riemannian manifold and $p_0$ a point in $M$. Let $U$ be a normal neighbourhood of $p_0$, image of $B_{\delta}(0) = \{ x \in T_{p_0} M : \left\lvert x \right\rvert < \delta \}$ ...
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Primitive of the matrix elements of irreducible representations of Lie groups

I am interested in the matrix coefficients $U_{ij}(g)$ of unitary irreducible representations of a Lie group $G$. In my case, these coefficients arise from the Peter-Weyl theorem. I would like to ...
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Do torsion and curvature have higher order analogues?

Consider the usual formulas for the Torsion and Curvature of an affine connection: $$T(X,Y)=\nabla_X Y -\nabla_Y X-[X,Y]$$ $$R(X,Y)=\nabla_X\nabla_Y-\nabla_Y\nabla_X-\nabla_{[X,Y]} $$ These formulas ...
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Constant Gauss curvature from bipolar projections.

Please help finding z-coordinate for constant positive and negative Gauss curvatures in Mongé form : $$ x= \sqrt{R^2 + T^2} + R \cos u ,\, y= R \sin u ,\, z= f(x,y). $$ $ R,T $ are constants. ...
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Integration Over Spherical Triangle and Change of Variable

Let $T$ denote a spherical triangle on the unit sphere, defined by vertices $u,v,$ and $w$. Let $\triangle$ denote a triangle. This could be the triangle defined by $u,v,w$ in $\mathbb{R}^3$ or the ...
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Buseman function and isometry in Cheeger-Gromell splitting proof

So I have the Busemann function $b^+$ as in the proof of the Cheeger-Gromell splitting theorem in Peterson and I want to show that if I have the isometry $f:(b^+)^{-1}(0)\times\mathbb{R}\rightarrow M$ ...
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Is there a way to compute the Poincaré dual of the following type of degree $(2n-2)$ de Rham class?

Given a closed, connected, symplectic manifold $(X^{2n},\omega)$, is there a systematic method to computing the Poincaré dual surface to degree $(2n-2)$ classes of the form $$[\omega]^{n-2}\cup B + ...
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$G$-invariant vector field coming from a principal bundle?

Let $\pi:P\longrightarrow M$ be a $G$-principal bundle. If $(U, \phi)$ is a local trivialization of this bundle then for every $x\in M$ we have a diffeomorphism $$\phi_x:P_{x}\longrightarrow G, ...
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Building Non-vanishing sections in a certain way

Assume you have a vector bundle $\Pi: E \rightarrow M$ where $E$ is the total space, $M$ is a compact manifold. Assume you know it is parallelizable. Let $\psi_i : \Pi^{-1}U_i \rightarrow U_i \times ...
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Solving PDEs given by a vector field

I have a non-compact smooth manifold $M$, a compactly supported smooth vector field $X$ on $M$ and a compactly supported smooth function $f$ on $M$. Does there exist a compactly supported ...
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Non commutative flows on the 2-sphere.

The title says everything, really. I'm looking for some flows on $S^2$ such that They do not commute. They are of some interest, or they are peculiar in some ways. Thanks
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Adjoint action on quotient space of Lie algebras and vector fields on quotient group

Let $G$ be a Lie group and $H$ a closed subgroup. Then $G/H$ has a unique structure of a smooth manifold with canonical projection $p: G \to G/H$. If $\mathfrak g = T_e(G), \mathfrak h = T_e(H)$ are ...
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Envelope of an Envelope

Background and Motivation Consider the following equation of family of ellipses in polar coordinates $$ r(\theta, \alpha ) = a\;\frac{- e \cos(\theta+ \alpha )+ \sqrt{ 1 - e^2 ...
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Where is $-2R_{ij}\nabla_jf$ from?

$M$ is a compact Riemannian manifold,$g_{ij}$ and $f(t)$ is defined as first picture. I want to compute (as equality with red line in second picture) $$ \int_M-\nabla_if\nabla_i(2\Delta ...
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Finding diffeomorphism given vector fields

Given a vector field how do you find the associated diffeomorphisms? Say I am given a vector field in Minkowski space, $\textrm{d}s^2 = -\textrm{d}t^2 + \textrm{d}x^2 + \textrm{d}y^2 + \textrm{d}z^2$, ...
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Upper bound for curvature of curve in $\mathbb{R}^2$

Suppose, $\gamma$ is a smooth curve in $\mathbb{R}^2$, and $\gamma$ lies inside some tangent circle $C$ to $\gamma$ at point $\gamma(0)$. I need to show, that $K_{\gamma}(0)\ge K_C$. I can solve this ...
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Weyl transformation of geodesic distance

Consider a Riemannian manifold $M$ with a metric $g$. For two points $x,y \in M$ the geodesic distance $d(x,y)$ is defined in the usual way. I would like to know if there is a formula expressing how ...
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Integral curves in null hypersurfaces

Let be $(M^{n+1},g)$ a spacetime (Lorentz manifold, connexe and time-oriented), $n\ge 2$, and $S\subset M$ a null hypersurface (codim $S=1$ and the restriction of $g$ to each tangent space $T_p ...
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Flat $G$-structures have Hol=Id

Exercise I've been given the task to show, given a flat $G$-structure, we have that $\text{Hol}=\text{Id}$ (here "Hol" is the holonomy group; furthermore a flat $G$-structure is defined be one such ...
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Complete manifold with non-positive sectional curvature

So I was told in a class that for a complete Riemannian manifold, $M$, with non-positive sectional curvature the exponential map at any point $p\in M$ is a covering map. Part of the proof just assumed ...
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Time dependent one-form change under Ricci flow

So this is part of a proof in Peter Topping's text, Lectures on Ricci Flow that I don't understand. Let $\delta A=-\text{tr}_{12}\nabla A$ and $\omega$ be a time dependent one-form. Also we use the ...
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variation metric tensor

$\newcommand{\Tr}{\operatorname{Tr}}$(I asked this on physicsexchange but no reply) I have the metric tensor $g_{\mu\nu}$. I want to make the variation of $\sqrt{-g}$ where $g=\det g_{\mu\nu}$. Can ...
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Find how a vector changes from parallel transport around a spherical triangle?

So I'm trying to find the components of a unit vector, initially on the sphere's equator parallel to the $\phi=0$ line, after parallel transport to $\phi=\phi_0$, then to the pole and then back to the ...
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What it means to “put together all the maps” here?

I'm reading Spivak's Mechanics book and he says the following when talking about Hamiltonian Mechanics Given a Lagrangian $L : TM\to \mathbb{R}$, at each point $a\in M$ the restriction $L_a = ...
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Show that a surface is minimal - scherk

Have to show that first scherk surface is minimal : We have $f(x,y)=(x,y,\ln(\cos x/\cos y))$ I am trying to show that using $H=1/2(k+k')=0$ (this is the definition given in the course) k and k' the ...
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Smooth of metric under Ricci flow

Let $(M,g)$ is a Riemannian manifold.$g$ evolve under $\partial_tg_{ij}(x,t)=-2R_{ij}(x,t)$. When I read Shi's derivative estimate, I need the metric $g_{ij}(x,t)$ to be continuous with respect to ...
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Constructing transition function of given manifold

This is extension of my previous question Moduli space about $CP^{N-1}$ and $T^* CP^{N-1}$., meaning of $\mathcal O(-1)$ in algebraic geometry? . What i have been considered are followings First ...
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Contraction of $(2k,2l)$tensor

In picture below ,how to get the equality in red box?
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Reference request for examples of integration of differential forms over manifolds

I am studying integration of differential forms over differentiable manifolds and I would like some reference where I can find examples of actual calculations illustrating the generalized Stokes' ...
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Can the curvature tensor (or any symmetric tensor) be diagonalized in a non-orthogonal basis?

I have a surface embedded in 3-space. It is parametrized using a set of two Gaussian coordinates, $x^1$ and $x^2$. From the parametrization, we may determine ${\bf a}_1$ and ${\bf a}_2$, the covariant ...
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Path Lengths on a Sphere example.

What does $\rho$ mean as to be measured along geodesics and more importantly how would i be able to parametrize this accordingly as being on the sphere's surface? I know that I have to use the First ...
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Compute of variation

How to compute the equality 2 ? I think it's to use normal coordinate,then $\Gamma_{ij}^k=0$, then $\nabla_h\Gamma_{ij}^k=0$, then $R_{ijk}^h=0$. Whether I am right ? $\delta$ is variation, ...
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Extend of cutoff function

$(M,g_t)$ is a family of Riemannian manifold ,$g_t$ evolve under Ricci flow $\partial_t g_{ij}=-2R_{ij}$. At $t=0$ ,we define $\varphi$ as below first picture . Then ,extend $\varphi=0, $ outside ...
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Tangent Spaces & Definition of Differentiation

What is meant by definition when we talk about the $c:(-\epsilon,\epsilon)\rightarrow M$ is that an interval on the curve? In differential geometry what is the difference between $D_{c(t)}$, ...
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relationship between curvature and exterior differential operator

I am reading selected topics in harmonic maps written by James Eells and Luc Lemaire. Let $\xi:V\to M$ be a vector bundle with connection $\triangledown$. In the book, author defines the curvature of ...
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Geometric Structures of a fixed area.

Lets $M_A$ be the space of metrics of area $A$ on a two dimensional surface $S$, and let $D_0$ be the group of area-preserving diffeomorphisms whose right action on $M_A$ is given by pullback. The ...
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Ruled surface swept out by normal lines of a curve

The problem is: "Say $\gamma:[a,b] \to \mathbb{R}^3$ is a curve of general type with principal normal vector field $\textbf{n} = t_2$. Show that the ruled surface $r(u,v) = \gamma(u) + v\textbf{n}(u)$ ...
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Show a inequality by maximum principle .

As below picture ,how to get the inequality 3 by inequality 1 and 2 ? It seem relate to PDE, but when I use the maximum principle , I don't know how to deal the $-F^2$ . Maybe, this question is ...
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Conformally invariant equations

I'm looking for conformally invariant equations in $\mathbb R^3$, and I can't find any other than the Laplace equation $\nabla^2 f=0$. Is this the only one?
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Find a parametrization of a plane curve if you`ve got its curvature function only

The curvature function is $\kappa(s)=\frac{1/b-1/a}{\sqrt{2}(1-\cos(s/b-s/a))^{1/2}}$ where $a<b$ are constant, and $s$ is the arc length of the curve. I know that the process in order to find ...
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Reference for the spectrum of the Bochner Laplacian on the 2-sphere.

I am looking for a reference for the spectrum of the Bochner Laplacian on $S^2$.
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Different notions of Submanifold

There are three types of submanifolds discussed in my book. Let $M$ be a smooth manifold. Then 1.) An immersed submanifold of $M$ is a set $S\subseteq M$ such that $S=F(S)$, where $F:N\to M$ is an ...
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intrinsic-ness of ordinary derivatives

The Laplace operator (or only the Laplace-Beltrami form of it?) is intrinsic: defined only in terms of the metric; does not reference the ambient space. The Laplace operator is a multidimensional ...