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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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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19 views

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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46 views

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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1answer
48 views

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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97 views

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
24 views

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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1answer
59 views

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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63 views

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 ...
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1answer
50 views

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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96 views

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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45 views

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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1answer
34 views

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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15 views

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
20 views

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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67 views

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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1answer
41 views

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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3answers
177 views

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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39 views

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
20 views

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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21 views

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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1answer
28 views

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 ...
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Confusion regarding terminology in Pressley's E.D.G

Here are two definitions taken from page $77$ of Pressley's Elementary Differential Geometry - $2$nd edition. Definition $4.2.1$ A surface patch $\sigma: U \to \Bbb R^3$ is called ...
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17 views

Singular points of the function F restricted to a torus generated by a circle without using a parametrization

The idea is to get the singular points of the function F: \begin{matrix} \bar{x}&=&x/\sqrt{x^2+y^2}\\ \bar{y}&=&y/\sqrt{x^2+y^2} \\ \bar{z} &=& z\end{matrix} restricted to ...
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1answer
22 views

Lie derivative of two differnt size related tensors

Let $\bar{M}=I\times M$ be a pseudo-Riemannian manifold equipped with metric $\bar g=-dt^2\oplus f^2g$ where $(M,g)$ is a Riemannian manifold, $I$ is an open connected interval and $f$ is a positive ...
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3answers
199 views

Prove that a manifold is not orientable

I have found a proposition who says: A manifold M is not orientable if it contains a Moebius band. How can I prove this?
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1answer
41 views

Affine manifolds which are *not* locally symmetric

Let $M$ be a manifold equipped with an affine connection $\nabla$. By standard theorems of differential geometry we have convex normal coordinate balls around every point in which geodesics are axis. ...
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35 views

Noncanonical isomorphism of spaces of differential forms

Let $\pi: V \to M$ be a smooth $n$-dimensional vector bundle over $M$. Are the spaces of differential forms $\Omega^i(V)$, $\Omega^i(V^*)$ noncanonically isomorphic? If so, how do I see this? Is there ...
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42 views

Two proof of Petersen's 'Manifold'

Picture below is from the 5 page of Petersen's Manifold. First, why diffeomorphism is forced to be a lieanr isomorphism? The define of diffeomorphism accords to Wiki. Second , what space the point ...
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Darboux theorem for symplectic manifold of degree 2

Given $p \in M$ and $\alpha \in \Omega^1(M)$ with $\alpha_p \neq 0$, show that there exists a neighbourhood $U$ of $p$ in $M$ and $f,g \in C^{\infty}(U)$ such that $\alpha|_U = f dg$. To show this ...
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Sequence of isometries converging to an isometry in Riemannian manifolds [closed]

Question: Let $\phi_n: M \to N$ be isometries between Riemannian manifolds such that $\phi_n$ converges uniformly to $\phi$ in the $C^0$ topology. Show that $\phi$ is a isometry. My problem is to ...
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Intuition about antisymmetrizing tensor equations

I was looking at the symmetries of the Riemann tensor, and tried to prove a couple of properties, namely If $\nabla$ is torsion-free, then: (i) $R^a_{\,[bcd]}=0$, and (ii) $R^a_{\,b[cd;e]}=0$. ...
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21 views

Metric evolving under Ricci flow with nonnegative scalar curvature is shrinking?

Let $g_{ij}(x,t)$ be a complete solution of the Ricci flow on a surface with bounded and nonnegative scalar curvature ,why the metric is shrinking ?
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1answer
44 views

Understanding wedge products for differential forms

I am trying to understand the derivation of coordinate expression for the Laplace-Beltrami operator (wiki here). The Wikipedia page says that $\nabla\cdot X$ is an operator mapping a function to a ...
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26 views

What is $P\times_G E$?

I know what is principal bundle and associated bundle according Wiki.But I am not understand what is $P\times_G E$ .Seemly it is bundle,but I am not sure what structure is it . Below picture is from ...
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22 views

Is this tensor contraction correct?

If I start off with the $\left(p,q\right)$-tensor given by $$T_{i_1,\dots,i_p}^{j_1,\dots,j_q}e_{i_1}\otimes\dots\otimes e_{i_p}\otimes\varepsilon^{j_1}\otimes\dots\otimes\varepsilon^{j_q}$$ and I ...
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2answers
34 views

A semisimple Lie group has no character; Am I right?

Let $G$ be a compact connected Lie group with semisimple Lie algebra ${\frak g}$. With the following reasoning, I show that there is no non-trivial Lie group homomorphism $$\chi:G\to S^1.$$ Is that ...
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39 views

Proving the Ricci identity

I'm trying to prove the Ricci identity Let $Z^a$ be a vector field, $R^a_{\,bcd}$ the Riemann curvature tensor and $\nabla$ a torsion-free connection. Then: ...
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Proof of an equation of Contact Riemannian metric structure.

Let $(M,g, \eta,\xi,\phi)$ be contact metric structure and $\{e_0=\xi,e_i,\phi e_i\}$ be a local orthonormal frame so-called $\phi$-basis. How to prove the following equation: ...
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2answers
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Isomorphism of space of horizontal vectors of $F\left(M\right)$

Let $\left(F\left(M\right),M,\pi\right)$ be the frame bundle of $M$. I am taking an element of $F\left(M\right)$ to be a pair $\left(p,u\right)$ where $p\in M$ and $u:\mathbb{R}^n\to T_pM$ is a linear ...
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38 views

Can a conformal map be turned into an isometry?

Let $f: (M, g) \to (M, g)$ be a conformal diffeomorphism of the riemannian manifold $(M, g)$, with $$ g(f(p))(Df(p) \cdot v_1, Df(p) \cdot v_2) = \mu^2(p) g(p)(v_1, v_2), \quad \forall p \in M, \, ...
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1answer
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Why is the image of the implicit function in the implicit function theorem not open?

We have a continuously differentiable function $f$ from $\mathbb{R}^{n+m}$ to $\mathbb{R}^n$, and we find a continuously differentiable function $g$ which maps points from $\mathbb{R}^m$ into ...
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Eigenfunction basis of Laplacian on a manifold

It is a well known result that for $\Omega$ bounded open set in $\mathbb{R}^n$, there exists a basis of $C^\infty$ eigenfunctions of the Laplacian for $L^2(\Omega)$. It is also known that there exists ...
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37 views

Volume enclosed by Implicit Surface

I am trying to calculate the Volume that's enlcosed by the surface: $$(x^2 + y^2 + z^2)^2 = xyz$$ The following is what i tried. I rewrote it in spherical coordinates where $x=r \cdot \sin\vartheta ...
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2answers
62 views

Frank Warner's definition of the Hodge star

Frank Warner's book, chapter 2, excercise 13 states the following: If $V$ is an oriented inner product space ($n$ dimensional) there is a linear map $\ast \colon \Lambda (V) \to \Lambda (V)$, ...
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Product of manifolds & orientability

I'm studying orientability of manifolds currently and I'm having trouble to prove the following: $M\times N$ is orientable iff $M$ and $N$ are orientable. I am able to prove that the product is ...
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24 views

Why does the Yamabe problem only consider compact manifolds

Why is the assumption of compactness so important to the statement of the Yamabe problem? What goes wrong if the manifold is not compact?