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Questions tagged [differential-geometry]

Differential geometry is the application of differential calculus in the setting of smooth manifolds (curves, surfaces and higher dimensional examples). Modern differential geometry focuses on "geometric structures" on such manifolds, such as bundles and connections; for questions not concerning such structures, use (differential-topology) instead. Use (symplectic-geometry), (riemannian-geometry), (complex-geometry), or (lie-groups) when more appropriate.

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Differentiable structure in $S^1$ such that the inclusion map is differentiable

I am asked to explicitly describe a differentiable structure on $S^1$ (with the usual subspace topology) such that the inclusion map $\iota: S^1 \to \mathbb{R}^2$ is differentiable. I've seen some ...
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Looking for a class name and a potential general solution

Is there a standard class name if any of the following differential equation there similar to Laplace equation and equivalent solution based on legendre polynomial, or spherical harmonics? $\partial^...
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Derivation of the Geodesic Equation from Hamilton-Jacobi Theory

I'm trying to prove a result in my general relativity class and I'm confused. If we have the Hamiltonian: $$ H = g^{\mu\nu}p_\mu p_\nu + m^2 $$ Subject to Hamilton's Equations: $$\frac{\partial x^\mu}{...
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Covering map of the universal cover $\widetilde{G} \rightarrow G$ for $G$ a Lie group is a homomorphism?

In a paper I'm reading, we have a compact Lie group $G$ and he says "We can identify $\pi_1(G)$ with the kernel of $\widetilde{G} \rightarrow G$. I can't seem to find anything that says that the ...
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How to calculate the geodesic curvature of a discrete 3D curve?

I have coordinates of a set of points that form a closed loop that lies in a 3D surface. I know the equation of the surface and I can calculate it's surface normal at any point. I found that for a ...
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Almost complex structure question

I know that if $M$ admits an almost complex structure $J$, then $\text{dim}_{\mathbb{R}}(M)=2k$, thus every odd-dimensional manifold doesn't admit an almost complex structure. My question is, are ...
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Tangent line of non-simple curves

I have a small confusion about curves. Given a non-simple parametric curve $\alpha : I \mapsto \mathbb{R}^{2}$. How does the tangent vector $T(s)$ and the tangent line $\{\alpha(s) + t T(s) : t \in \...
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Curvature forms as exterior covariant derivative?

I have read on several forums like this one, that given a connection form $\omega$ on a principal bundle and its curvature form $\Omega$, I can state that $\Omega=d_\omega\omega$ alike I do in the ...
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Partial Derivatives - Covectors - Antisymmetric Tensors

Suppose that $V_{\mu} (x)$ transforms like a covariant vector under a general curvilinear coordinate transformation $x'^{\nu} = x'^{\nu} (x^{\mu})$ . Show that the partial derivatives $V_{\mu,\nu} \...
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if a tangent at a point of a manifold taken as a function is zero at two points, it is zero at all points

Let $M$ be a manifold and $m\in M$. We call $t$ a tangent at $m$ if for every pair $(f,g)$ of smooth real functions defined in a neighborhood of $m$ and for every pair $(a,b)$ of real numbers, we have ...
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How to find an embedded submanifold in $\mathbb{R}^{n}$ which pass through some given points in $\mathbb{R}^{n}$ and has the lowest dimension?

How to find an embedded submanifold in $\mathbb{R}^{n}$ which pass through some given points in $\mathbb{R}^{n}$ and has the lowest dimension? I have no idea with this question. I know, at least, the ...
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Question on Solid Angles and Linking Number

In Spivak's Calculus on Manifolds there is a question about solid angles and linking number that confuses me. Suppose $f([0,1])=\partial M$ where $f$ is a closed curve and $M$ is a compact oriented ...
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Computing all possible conformal factors on the sphere

Proposition to prove. Let $\tau\colon \mathbb S^n\to \mathbb S^n$ be a conformal map, meaning that $\tau^\star g= \Lambda^2 g$ for a scalar field $\Lambda$. (Here $g$ denotes the standard metric ...
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Integration on Differential Forms.

So I know that divergence and curl of a vector field $F$ can be related to a differential form $\alpha$ by div $ F = \star d \star \alpha$ and curl $ \cdot F = \star d \alpha $ , where $\star$ is the ...
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Surface Area of generalized polar parametrization

Suppose we have a smooth domain $D$ in $\mathbb{R}^n$, which can be parametrized by polar coordinates $(\zeta, r(\zeta))$. Here, $\zeta$ denotes a point on the $(n-1)$ dimensional unit sphere $B$ and $...
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Existence of a countable locally finite cover with nonempty intersection of two adjacent elements

Let $\Omega$ be an open connected set in $\mathbb{C}$, not necessarily bounded. Does there exist a countable locally finite cover of $\Omega$ consisting of only open discs $\{ B(z_i, r_i): i\geq 1\}$ ...
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Phase, Isochrons, Isochrons map and Lift

at the moment i read the following paper: https://arxiv.org/pdf/1512.04436v1.pdf I have some questions about it and i hope someone can help me. On page 4/5 they introduce isochrons and the isochron ...
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Questions on Cartan's magic formula $\mathcal{L}_X=i_X \circ d + d\circ i_X$

Algebra $A$ is called graded algebra if it has a direct sum decomposition $A=\bigoplus_{k\in\Bbb Z} A^k$ s.t. product satisfies $(A^k)(A^l)\subseteq(A^{k+l}) \text{ for each } k, l.$ A ...
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Circle inscribed between two curves

Consider the plane region $S_n$ bounded from above and below for the graphs of $f_n(x)=x^{1/n}$ and $g_n(x)=x^n$, $0\le x\le1$. How to find the radious and center of the circle inscribed in $S_n$? ...
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Do Carmo Section 2.2 Proposition 2 Proof Clarification

I am having some difficulties in understanding the proof given by Do Carmo of the following proposition. I will run through the proof and comment on it as it progresses and place my questions ...
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What intrinsic property determines whether a function is analytic

Given we know the value of all order derivatives at a point $x_0$ for a given f(x). As per my knowledge all the geometric properties like slope, curvature, convexity are functions of solely the ...
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Is the warped product: $dr^{2} + r^{2} \gamma$ always flat in 3 dimensions?

Suppose you have a Riemannian 2-sphere $(S^{2} , \gamma)$. Define a metric $g$ on $M = S^{2} \times [1,\infty)$ in this way: If $(x_{1}, x_2)$ is a coordinate chart on $S^{2}$ then $$g = dr^{2} + r^{...
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Why is true this equality? Afinne conection

Let $\gamma : [0,l] \to M$ be the only geodesic joining $p$ and $q$ in a complete riemannian manifold, where $q$ is out of cut locus of $p$.Let $E_1,...,E_n = \gamma'$ a orthonormal parallel field ...
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How to prove that the equation is not possible

I came across another very complex equation (calculating the Gaussian curvature of a surface): \begin{align*}\frac{-m}{2}=&(2A^2+A)(Du+k)^3u^{(3+6A+4B)}\\ &+AD(Du+k)^2u^{(6A+4B+4)}\\ &+(...
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Finding curvature of a surface of revolution with metric inherited from $\mathbb{R^3}$

I have a curve in $\mathbb{R^3}$ as $z= f(x)$ in the $xz$- plane and we let $S$ be the surface generated from this curve in the space by revolving it about the $z$- axis. Now, I'm asked to find it's ...
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Weitzenböck identity for $TM$-valued differential forms

Let $M$ be a Riemannian manifold, and let $\nabla$ denote its Levi-Civita connection. We have two second order differential operators $\Gamma(T^*M \otimes TM) \to \Gamma(T^*M \otimes TM)$: The ...
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Geodesics under coordinate transformation

Consider components of metric tensor $g'$ in a coordinate system $$g'= \begin{pmatrix} xy & 1 \\ 1 & xy \\ \end{pmatrix} $$ We can transformation rule which brings $g'$ to euclidean metric $...
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The existence of a smooth vector field locally

Suppose we have a smooth $k$-dim manifold $M \subset \mathbb{R}^𝑛$ and the tangent space at every point $p \in M$ (Here we translate every tangent space passing the origin point) has non-zero ...
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Are real linear maps of smooth sections locally determined?

Let $ \pi_{ 1 } \colon E_{ 1 } \to M $ and $ \pi_{ 2 } \colon E_{ 2 } \to M $ be smooth vector bundles (of finite rank) over a smooth manifold $ M $, and consider a map $ T \colon \Gamma ( E_{ 1 } ) \...
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Prove that $\alpha(t)$ lies on a sphere given that $\alpha(t)-a$ is orthogonal to T(t) [duplicate]

Let $\alpha(t)$ be a regular curve. Suppose there is a point a in R^3 such that $\alpha(t)-a$ is orthogonal to T(t) for all t. Prove that $\alpha(t)$ lies on a sphere. So we know $<\alpha(t)$ - ...
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Exterior derivative of a 1-form on a surface for non-regular mappings

I would appreciate some help for this problem. I have no idea how to start. Let $M\subset \mathbb{R}^3$ be a smooth surface. Let $\phi$ be a $1$-form on $M$. By definition, the exterior derivative of ...
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Tangent Map of an Isometry and the Shape Operator

I have spent a lot of time on this problem and would appreciate some help. Please bear with me. Let $M$ be a surface and $F\colon\mathbb{R}^3\rightarrow\mathbb{R}^3$ an isometry. Denote with $F_\...
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Finding if there is a diffeomorphism which pushes forward one vector field to the other

Given a pair of vector fields on a real line how can I see if there exists a diffeomorphism, which pushes forward one to the other? I have $$ V_1 = 2sinx\frac{\partial}{\partial x}, \enspace V_2 = sin^...
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Question related to orientation on a arbitrary oriented manifold

This is a section from Loring Tu's book Introduction to Manifolds page 244 Second Edition. My question is as follows: Towards the end of the text in the image he says that an oriented manifold can be ...
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If $f_n:\Omega\to\Omega$ are homeomorphisms of a planar domain $\Omega$ such that $f_n\to f$, $f_n^{-1}\to g$ in $L^1$, is $f=g^{-1}$?

In Marchioro and Pulvirenti's book Mathematical Theory of Incompressible Nonviscous Fluids, the proof of global well-posedness of the 2D Euler equation in a bounded domain $\Omega\subset\mathbb R^2$ ...
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Condition for Riemannian distance to be equal to metric distance

If $M$ is a metric space than it is a topological space and if it is locally homeomorphic to to $R^n$ we say that it is a manifold and if we equipped this manifold with a inner product $g_p$ on ...
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Can someone help in computing curvature tensor of a surface?

I have a problem, I've been thinking about all day. Came across this while browsing some lecture notes online. So, I have a surface in space say, described as $z= f(x,y)$ and I want to find it's ...
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Doubt about curves with the same curvatures

I am having some trouble with this theorem: In Klingenberg's "A course in differential geometry" on page 13 it claims: Let $c_1, c_2: I \to \mathbb{R}^n$ be two curves ($C^{\infty}$) ($n-1$)-...
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The shrinking sphere example

I am having some issues with the vocabulary employed by authors when they refer to some solutions of the Ricci flow equation. For instance, the shrinking sphere example. It seems odd to me when I ...
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2answers
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Proving a Line-integration along a parametrized curve identitiy.

(this question were asked after studying line integrals) 1- Show that if $C$ is the graph of $y=f(x)$, $a \leq x \leq b,$ and if $F$ is a function of 2 variables defined on C, then $$\int_{C} F(x,y)...
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Connected boundary implies $\pi_1(M,\partial M)=0$.

I have two questions: Let $M$ be a compact connected manifold with boundary. 1, If the boundary $\partial\tilde{M} $ of universal covering $\tilde{M}$ is connected, is $\partial M$ connected? How ...
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Tangent vectors as equivalence classes of triples and ordinary vectors

I am using this document as a reference on tangent spaces etc. In the section on tangent spaces, the author provides three equivalent definitions of a tangent vector, the first being the intuitive ...
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Relation between Normal coordinates and Cartesian coordinates in flat space

Let $\gamma_v$ be the unique maximal geodesic with initial conditions $\gamma_v(0)=p$ and $\gamma_v'(0)=v$ then the exponential map is defined by $$exp_p(v)=\gamma_v(1)$$ If we pick any orthonormal ...
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Proving a that $C_c(TM)$ injects into $C_0(T^*M)$ through Fourier Transform

Let $M$ be a smooth manifold. Equip $TM$ with an Riemannian structure. We let $f \in C_c(TM)$ and define a homorphism into $C_0(T^*M)$ by $$ (x,w) \in T^*M, \hat{f}(x,w) = (2\pi)^{-\frac{n}{2}} \...
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Global Kodaira correspondence for analytic families of compact complex manifolds

First of a necessary definition. An analytic family of compact complex submanifolds of a complex manifold $Z$ with parameter space $M$, which is a complex manifold, is a complex submanifold $F\subset ...
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1answer
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How can we apply generalized Stokes' theorem to a non-oriented manifold with boundary?

I do not really know much about the boundary of non-oriented manifold. A boundary of oriented manifold, if it exists, has a sign. If you reverse the orientation, the boundary picks up an extra ...
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Tangent vector as velocity of curve that is in tangent space of manifold but not in the tangent space of submanifold.

This is the problem 5-19 from the second edition of Lee's "Introduction to Smooth Manifolds" (see here). I need to see clear why is false if S is not embedded. I already know that the eight-figure ...
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Is the differential at a regular point, a vector space isomorphism of tangent spaces, also a diffeomorphism of tangent spaces as manifolds?

Note: My question is not "If $f$ is a diffeomorphism, then is the differential $D_qf$ an isomorphism?" My book is From Calculus to Cohomology by Ib Madsen and Jørgen Tornehave. I didn't study much of ...
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1answer
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Understanding the meaning of immersion for Manifolds

Let $f: X \rightarrow Y$ be a smooth map of manifolds. $f$ is an immersion at $x \in X$ if $df_x: T_x(X) \rightarrow T_y(Y)$ is an injective map where $y = f(x).$ $T_x(X)$ is the tangent plane of $X$ ...