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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Definition of a smooth function between surfaces

If $S_1, S_2 \subset \Bbb R^3$ are two smooth surfaces, then what is the formal definition of a smooth map from $S_1$ to $S_2$? I am studying from Pressley's EDG, and the definition is given only in ...
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explaination of the metric tensor on another manifold?

In skew -product decomposition the following features are observed :- 1.the Riemannian Manifold $(M,g)$ has a product form of $$M=R\times \Theta$$ Where $\Theta ,R $ are connected $C^\infty$ ...
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Canonical notion of parallel transport

I have a "What is the right search term?" style question: Suppose $S\subset\mathbb{R}^3$ is a surface and that we are given two points $x,y\in S$. Furthermore, take $v_x\in T_x S$ to be a tangent ...
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Correspondence between vector bundles and locally free sheaves

I am trying to look at smooth manifolds in the context of locally ringed spaces. Vector bundles have a characterization in terms of sheaves of modules: if $X, \mathcal{O}_X$ is a topological (or ...
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Every manifold admits a vector field with only finitely many zeros

Let $M$ be a smooth manifold. I am trying to prove that $M$ admits a vector field with only finitely many zeros. This will follow if we can find a function $f : M\rightarrow \mathbb R$ such that ...
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Codifferential and Hodge star

Is this true, \begin{align} \notag \delta (f * \Omega )= f \delta (*\Omega)? \end{align} $\delta$ denotes codifferential, f is a function, $\Omega$ is a k-form and * is a Hodge star operator.
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“Reparameterization principle” and the way it is used

I am reading from Pressley's Elementary Differential Geometry. On page $79$, the author states a principle and says that it will be used throughout the rest of the book. Statement: The principle ...
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524 views

How does the boundary property usually work in PDE?

This may be related to the question Is the hypersurface of class $C^k$ a $C^k$-differentiable manifold?. In almost every chapter of a PDE textbook(e.g. Folland's Introduction to Partial Differential ...
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Is $f:SO(n)\rightarrow S^{n-1}$, $f(A)=(A^n_i)_i$ a submersion?

Let $f:SO(n)\rightarrow S^{n-1}$, $f(A)=(A^n_i)_i$, that is $f(A)$ is the last row of $A$. Show that $f$ is a submersion. I'm not sure how to calculate $df$, because I only know how to calculate the ...
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any open set in $\mathbb{R}^n$ is a $n$ dimensional manifold

I am trying to show this using the definition: M is a k-dimsensional submanifold of $\mathbb{R^n}$ if for all $x \in M$ the following condition holds: There exists an open set $U \subset ...
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Gaussian curvature in polar coordinates

Find the expression for the Gauss curvature in the polar coordinates associated to the exponential map. I thought about using Gauss's lemma: if $(r,\theta)$ are polar coordinates in the tangent plane ...
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geodesic on an ellipsoid

Find all the geodesics which pass through the point $(a, 0, 0)$ on the ellipsoid $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$. What parametrization can i use to get the first fundamental form? ...
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Computing of proof of Li-Yau estimate

I try to compute the red line in picture below: \begin{align} \Delta(\partial_tL) +R\Delta L +\partial_t R &=\Delta(Q+|\nabla L|^2)+R(\frac{\Delta R}{R}-\frac{|\nabla R|^2}{R^2}) + \Delta R +R^2 ...
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Calculating the Lie algebra of $SO(2,1)$

I am trying to calculate the Lie algebra of the group $SO(2,1)$, realized as $$SO(2,1)=\{X\in \operatorname{Mat}_3(\mathbb{R}) \,|\, X^t\eta X=\eta, \det(X)=1\},$$ where $$\eta = \left ( ...
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finding an integral manifold of a distribution

I have vector fields $\begin{cases}X_1 &=& -y\frac{\partial}{\partial x}+x\frac{\partial}{\partial y},\\ Y &=& (x-y)z\frac{\partial}{\partial x} + (x+y)z \frac{\partial}{\partial y} + ...
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2answers
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What does $C^k$ at a single point mean?

I'm reading textbooks on manifolds. I usually see a saying that "function $f$ is $C^k$ at a point $p \in M$". I am confused with this. What does this mean? Furthermore if $f$ is $C^k$ at $p$, is it ...
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Chain rule for general manifolds

So, I need an explanation why shall it be $\frac{d}{dt}|_{t=0}(f(\phi_X^t(m)) = d_m f \frac{d}{dt}\phi_X^t(m)|_{t=0}$ where $\phi_X^t$ is a flow, m is point in a given differentiable manifold M. I ...
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Why do we need the $S \otimes L^{1/2}$ bundle product to determine a $Spin_c$ structure?

I am reading Marino's book on topological field theory and 4-manifolds and I am very confused in the construction of the $Spin_c$ structures for manifolds that do not admit a $Spin$ structure. In the ...
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Construct a lifting from the the total space of a fiber bundle to its spoace of 1-jets

I am reading the book "Convex Integrtion Theory by D.Spring" and need some help. Let $\pi:X \rightarrow V$ be a smooth fiber bundle over a smooth base manifold $V$. Let $h\in\Gamma^{0}(X)$ and ...
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Relationship between invariant and harmonic forms

Let $G$ be a compact real Lie group which is also a complex manifold (note that I do not want $G$ to be a complex Lie group). Let $\Omega^{p , q}(G)$ denote the space of smooth $(p , q)$-forms on $G$, ...
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Euler classes of oriented $2$-dimensional vector bundle, oriented $S^1$-bundle same?

As the question title suggests, are the Euler classes of an oriented $2$-dimensional vector bundle and of an oriented $S^1$-bundle the same?
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Orthogonality of ruling and directrix of a ruled surface

Let $M$ be a ruled surface of $\mathbb{R}^3$ with a regular parametrization given by: $$x(u,v)= \alpha(u) + v\beta(u)$$ where $\alpha' \neq 0$ and $ ||\beta || = 1$. I want to show that ...
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Proving a curve is generalized helix

Curve is defined in a following way $\alpha(t)=(2t,\ln t,t^2)$. I want to prove that it is generalized helix. I tried parametrizing through arc length but it got messy fast. Any ideas on how to ...
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How to graphically represent $\ddot x$?

We know that given a differential equation: $$\dot x = f(x), x \in X$$ The $\dot x$ is understood as the tangent vector on the solution trajectory $x$ lying in the tangent space of $X$ What about ...
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87 views

How to prove that the wedge product is the determinant (Spivak's Claim)

In the book I am using now, Spivak's A comprehensive introduction to differential geometry volume 1 I have a question on page 205. Because he says the following I think this is equivalent to say the ...
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finding geodesics on the surface $z=x^2$

Find all the geodesics on the surface $z=x^2$. I found the metric and the Christoffel symbols but i do not know what to do next, any hint ?
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symmetric and alternating tensors in differential geometry

The following is an excerpt from Chern's Lectures on Differential Geometry: I don't see how the proof shows the other direction of the set inclusion. Would anybody explain the logic in the ...
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In proof of tangent space being a plane

can someone explain why this is true? If $\sigma$ is a surface patch of a surface $S$ and $p$ is a point on the image of $\sigma$ and if $p$ lies in the image of a curve $\gamma$ contained in $S$ say ...
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What is skew-product decomposition?

What is skew-product decomposition of Brownian motions referring to this paper Pauwels ,Rogers
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What is the horizontal space of trivial hermitian line bundle?

Suppose $L=M\times\Bbb C$ is a trivial holomorphic line bundle on a complex manifold $M$, and suppose there is a Hermitian fibre metric $h$ on $L$. Question: What is the horizontal space of ...
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Divergence of vector field on manifold

This is a follow-up question to the one I made here. On the wiki page, the divergence of a vector field $X$, denoted $\nabla\cdot X$, is defined as the function satisfying $\left(\nabla\cdot ...
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Codimension one foliation

let $f:M \rightarrow \mathbb{R}$ be a nowhere vanishing function defined on a Riemannian manifold $(M,g)$. consider the distribution $\ker(df)$. This is clearly involutive and thus defines a ...
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1answer
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Unit speed spherical curve curvature [closed]

I want to prove that unit speed spherical curve $\beta$ satisfies following inequality $$\kappa_{\beta}(s)\geqslant \frac1{R},$$ where $\kappa$ is curvature and $R$ is the radius of the sphere, that ...
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Short exact sequences from the Euler sequence.

I was reading an article in which the author said that the sequence $\require{AMScd}$ \begin{CD} 0 @>>>\Omega ^1_{\mathbb{P}^n} @>>> \mathcal{O}_{\mathbb{P}^n}(-1)^{\oplus n} ...
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Derivative of vector field on a manifold sends the tangent plane to itself

A vector field $\vec{v}$ on a (smooth) manifold $X \subset \mathbb{R}^N$ is a map $\vec{v}:X \to \mathbb{R}^N$ such that $\vec{v}(x)\in T_x(X)$ for every $x \in X$. Suppose $\vec{v}(x)=0$, show that ...
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The connection between differential forms and ODE

Is there a connection between being an exact differential equation and being an exact differential form? I always found it bothersome with basic ode that you could somehow treat dy/dx as a bona fide ...
2
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1answer
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Area of Mobius strip

I want that to give a meaning to the notion of area for Mobius strip. I know that Mobius band is nonorientable surface. How can I set up an integral to compute it? What's your idea for the following ...
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Geometrical Interpretation of Tensors (intuition)

How would one describe to a non-mathematician (an undergraduate physicist actually- so please do use mathematics-i am not even sure if they can be described without mathematics!) what do tensors ...
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Example of exact form

Consider the differential 1-form $\omega = ydx+dy$. I need to show that this is not exact, and find an example of a function $G(x,y)$ such that $G\omega=G(x,y)(ydx+dy)$ is an exact form. I have done ...
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What is “the line bundle $\Omega^n(M)$”?

In this Wikipedia article here what is "the line bundle $\Omega^n(M)$"? It seems to me that there can be many different line bundles on a smooth manifold $M$ so it's not clear to me what ...
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Compute the characteristic strip and conoid solution of a geometric PDE

This is a problem from Fritz John's Partial Differential Equations, which I'm working through for self-study. Given a family of spheres of radius $1$ with centers in the $xy$-plane ...
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1answer
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Show that exist a unique field $G$ on $TM$ whose paths are of the form $t\to (\gamma(t),\gamma'(t))$, where $\gamma$ is a geodesic on $M$.

Show that exist a unique field $G$ on $TM$ whose paths are of the form $t\to (\gamma(t),\gamma'(t))$, where $\gamma$ is a geodesic on $M$. My approach: Suppose such field actually exist, consider a ...
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Property on the derivative of a wedge product of two n-forms

I'm trying to prove the following property of $n$-forms. When $w_1$ is a $n_1$-form on $M$, $w_2$ a $n_2$-form also on $M$, and $d$ denotes the exterior derivative $$\require{cancel} d(w_1\wedge ...
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Does anyone know a book on sketching surfaces?

Is anyone aware of books or sources dedicated to sketching surfaces? Sort of like Forst's An Elementary Treatise on Curve Tracing, but on surfaces; or a book that has a fair few chapters dedicated to ...
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Example of a disconnected manifold where the tangent space is not the dimension of the manifold?

Wikipedia says that the tangent spaces of a connected manifold all have the same dimension, equal to that of the manifold. Well, is there an example of a simple disconnected manifold that doesn't ...
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Computing the total curvature

Let $C$ be the curve in $\Bbb{R}^2$ given by $(t-\sin t,1-\cos t)$ for $0 \le t \le 2 \pi$. I want to find the total curvature of $C$. I found it brutally by finding the curvature $k(t)$, and then ...
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Flow of vector fields

The flow of a vector field is in general not globally defined. I know that, in case the manifold is compact, or the vector field is compactly supported, its flow exists globally. What can one say ...
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1answer
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Non-existence of $1$-dimensional tangent distributions on the sphere $S^2$

I am reading about geometric quantization and real polarizations, and it is claimed that there exist no real polarizations on the sphere $S^2 \subset \Bbb R^3$ because every tangent field on $S^2$ ...
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Dirichlet problem for a ball in a Riemannian Manifold

I'm trying to work through the 2nd chapter "Volume comparison" of Jeff Cheeger's "Degeneration of Riemannian Metrics under Ricci Curvature Bounds". But with the last section I have some problems. ...
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Can we have a diffeomorphism from a subset of $\Bbb R^2$ into a subset of $\Bbb R^3$?

In a lecture, our professor defined an allowable surface patch for a surface $S \subset \Bbb R^3$ to be a diffeomorphic surface patch of $S$. But is is possible to have a diffeomorphism between an ...