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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Why is the 'line-element' non-integrable?

I'm reading The Variational Principles of Mechanics by Cornelius Lanczos; here is the concerned excerpt: A second landmark is the geometry of Riemann, which grew out of the ingenious ...
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29 views

Such a manifold is homeomorphic to a sphere

I recently read that if a compact differentiable manifold admits a real function with only two critical points, then it is homeomorphic to a sphere. If the function is Morse, this follows from ...
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21 views

Choice of a horizontal tangent space of a principal bundle

Let $\pi:P\to M$ be a principal bundle with group $G=\pi^{-1}(p)$, and let $u\in P$ and $p=\pi(u)$. As I understand it, the choice of the vertical tangent space $V_uP=\mathrm{ker}(\pi_*)$ is natural, ...
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19 views

Invariance of the rank of the trace of Riemannian curvature under a change of frame

Let $$ R=\begin{pmatrix} R_{11} & ... & R_{1n} \\ &...\\ R_{n1} &...& R_{nn} \end{pmatrix}, $$ where $R_{ikjl}$ is curvature tensor of a Riemannian manifold $(M,g)$ and ...
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20 views

Area of a complete, simply-connected surface with non-positive Gauss curvature is infinite

I am reading an article of F. Xavier about the Gauss map of complete, non-flat minimal surfaces in $\mathbb{R}^3$ (reference: http://www.jstor.org/stable/1971139?seq=1#page_scan_tab_contents). In the ...
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33 views

How can I prove that interior product obeys a graded Leibniz rule?

I want to prove that $i_{X}(\omega\wedge\phi)=i_{X}\omega\wedge\phi+(-1)^{k}\omega\wedge i_{X}\phi.$ I was thinking I many be able to adapt the proof that the exterior derivative obeys the graded ...
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42 views

Question about “Stochastic Analysis on Manifolds”

After Definition 2.3.1 Hsu says that if $M$ is a closed submanifold of $\mathbb{R}^N$ then a semimartingale $X$ on $M\subseteq\mathbb{R}^N$ should satisfy $$X_t=X_0+\int_0^tP\left(X_s\right)\circ ...
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23 views

If a metric tensor is not conformally equivalent to the flat metric

If on a manifold $M$ we have two metrics $g_{ab}$ and $g'_{ab},$ which are not conformally equivalent, and we say that $(M,g_{ab})$ is a flat manifold, does it follow that $(M,g'_{ab})$ is not flat? ...
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54 views

Gradient and Divergence in Riemannian Manifold

Let $M$ a riemannian manifold. Let $X\in\chi(M)$ and $f$ a function $C^{\infty}$ in $M$. Define the divergence of $X$ as a function $div X:M\to\mathbb{R}$ given by $divX(p)=\mbox{trace of the linear ...
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46 views

The orbit of a compact Lie group action

Let $G$ be a compact Lie group acting on a manifold $M$. For each $p\in M$, we define the orbit of $p$ as $G\cdot p:=\{g\cdot p: g\in G\}$. The isotropy group of $p$ is $G_{p}=\{g\in G:g\cdot p = ...
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49 views

What defines a fiber bundle?

I am slightly confused by Geometry, Topology and Physics by M. Nakahara. The following definition of fibre bundle $(E, \pi, M, F, G)$ is given: $E, M, F$ are differentiable manifolds, $E$ being the ...
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1answer
72 views

Pushforward of a vector

The push forward of vectors allows to transform the components of a vector $X$ to be pushed along a map $h:\mathcal{M}\to\mathcal{N}$ between the manifolds $\mathcal{M}$ and $\mathcal{N}$. This is ...
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30 views

A Riemannian metric on the torus $T^n$

This exercise is from Do Carmo, Riemannian Geometry. Introduce a Riemannian metric on the torus $T^n$ in such a way that the natural projection $\pi:\mathbb{R}^n\to T^n$ given by ...
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32 views

Bott&Tu Definition: "Types of Forms:

In Bott&Tu's well-known book "Differential forms in Algebraic topology", they note -(p34): every form on $\mathbb{R}^n \times \mathbb{R}$ can be decomposed uniquely as a linear combination of two ...
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1answer
38 views

Diffeomorphism maps geodesics to geodesics

Let $f:M \to N$ a diffeomorphism between riemannian manifolds of the same dimension. What are sufficient conditions for $f$ to map geodesics to geodesics? Of course, if $f$ is an isometry this occurs, ...
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31 views

Two definitions of the first Chern class

there are two definitions of the first Chern class that I don't know how to relate - hints and references are both welcome. So, first approach: say that I have a complex vector bundle $E\to M$; I can ...
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27 views

Christoffel symbol and geodesics [closed]

In a proof of some theorem I came across, stays that since $\Gamma_{11}^{2}=0$ certain (asymptotic) curve on surface has vanishing geodesic curvature. How did they got it? Beltrami's equation or ...
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29 views

Total Variation and geodesic Curvature

If $f:[0,1]\rightarrow \mathbb{R}^d$ is a smooth curve, then is there are relationship between the total variation of $f$ and the geodesic curvature of $f$? I expect they both should be zero iff $f$ ...
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11 views

Gaussian blur over (or random walk in) a surface mesh

Let $V$ be the set of mesh vertices, connected by edges $E$, forming a mesh that represents a surface embedded in $\mathbb{R}^3$. On this mesh a function $f:V\rightarrow\mathbb{R}$ is defined. For ...
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1answer
66 views

How to determine A?

I'm struggling with the following problem: Let $(\gamma, \tau)$ be an arc-length parametrized curve and $\mathcal A: \mathbb R^2 \to \mathbb R^2$ be a Euclidean transformation so that the curve $\hat ...
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1answer
25 views

Gauss application is an isometry

My question is very simple : What can we say of a compact surface $\mathcal{S} \subset \mathbb{R}^3$ satisfying that the Gauss application $N$ is an isometry : $\mathcal{S} \to \mathbb{S}^2$ ...
5
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61 views

Tangent bundle of sphere as a complex manifold

I'm trying to show that the tangent bundle, $TS^n$ of the n-sphere $S^n$ is diffeomorphic to the set $\sum z_i^2 = 1$ in $\mathbb{C}^{n+1}$. It's relatively straightforward to see that the tangent ...
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1answer
49 views

Coordinate charts vs. coordinates on manifolds

I just realised that I'm confused what coordinates really means in the context of manifolds. For example, say $M=S^2$. Then we can define smooth charts as follows: Let the open sets be $U = S^2$ ...
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34 views

Find the volume(solid) , transform rectangular function to polar function

if want find volume of this problem Under $ f(x)=x^2+y^2-4 $ and inside $ x^2 + y^2=9$ in plane $z=0$ Can I use this integration in polar functions? $$\int_0^{2\pi} \int_2^{3} (r^2 - 4) r ...
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1answer
23 views

Geometric meaning of the contact condition?

I am trying to understand contact structures. The definition of a contact manifold is this: Let $M$ be a $2n + 1$-manifold and let $\omega$ be a differential $1$-form such that $\omega \wedge ...
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1answer
35 views

Does every $C^1$ closed differential form differ from some $C^\infty$ closed form by an exact form?

Question: On a $C^\infty$ manifold, does every $C^1$ closed differential form differ from some $C^\infty$ closed form by an exact form? Motivation: This result holds for $C^1$ closed 1-forms on a ...
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22 views

Normal vector field for an immersion

I know that a hypersurface $M$ of a riemannian manifold $N$ is orientable iff there exists a globally defined unit normal vector field $\eta : M \to TM^{\perp} \subset TN|_M$. Does the same hold for ...
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12 views

Writing hyperplane as a graph

I have a geometry question. If S is a hypersurface of dim n-1, by moving any point $x_0$ to the origin, and making the tangent plane of $x_0$ the hyperplane $x_n=0$, can we always find a function f ...
2
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40 views

O'Neill's differential geometry: typo in formula for partial derivative?

I am working through Barrett O'Neill's Elementary Differential Geometry and I'm mildly confused. Exercise 3 in section 4.3 ask you to verify that $$\mathbf{y}_{u}=\mathbf{x}_{u}\frac{\partial ...
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2answers
46 views

Formula of a two form for a parallelizable manifold

Let $M^n$ be a parallelizable manifold with the nowhere dependent vector fields $X_1,\ldots, X_n$ forming a basis for the tangent space at each point of $M$. The Lie brackets of these fields are ...
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1answer
53 views

Deciding whether a form in the exterior power $\bigwedge^k V$ is decomposable

Let $V$ be a vector space and $\bigwedge^kV$ be the $k$th exterior power. I'm trying to find a condition that characterizes when an element $\omega \in \bigwedge^kV$ is decomposable in the sense that ...
2
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1answer
34 views

Show that a k-form can be expressed as wedge product

I'm trying to show that given a $1$-form $\omega$ and a $k$-form $\alpha$ such that $\alpha \wedge \omega = 0$ then there exists a $(k-1)$-form $\beta$ such that $\alpha = \omega \wedge \beta$. I'm ...
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12 views

Metric with scalar curvature-1

i have this simple question : can someone give me the definition or an indication to know what is the "Metric with scalar curvature-1 " thanks .....
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23 views

Easier way to induce an orientation to the border of a manifold

I'm working in the following exercise: Let $M=\{(x, y, z): x²+y²=1 \,\text{and}\, 0\leq z \leq 1\}$. Let $\alpha:(0,1)²\rightarrow \mathbb R³$ be given by $\alpha(u, v)=(\cos u, \sin u, v)$. ...
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1answer
35 views

Is a translation in a compact Lie group homotopic to the identity?

The following exercise is from Guillemin and Pollack, Differential Topology. Show that the Euler characteristic of the orthogonal group (or any compact Lie group for that matter) is zero. Hint: ...
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242 views

The quotient of a manifold by a submanifold is never a manifold?

Let $M$ be a connected smooth manifold. Let $S$ be a connected embedded submanifold of positive dimension and co-dimension, which is also a closed subset of $M$. Is it true that the quotient space ...
2
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23 views

Raising index on covariant derivative

So suppose $X$ is some vector field and $t$ is a tangent vector to some curve on some smooth manifold. Then $t^a\nabla_a X$ gives the directional derivative of the vector field in the direction of ...
2
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1answer
18 views

Showing that, at an elliptic point, a surface lies on one side of the tangent plane.

Let $p\in S$ be an elliptic point of a surface $S$. I want to show that there exists a neighbourhhod $V$ of $p$ in $S$ such that all points in $V$ belong to the same side of the tangent plane ...
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27 views

On Automorphism on Space of Smooth Functions

Given a Operator $T$ (an Automorphism) on the subspace $X$ of Smooth functions on $\mathbb R ^n$, $\mathcal C^\infty(\mathbb R^n)$ $$ X=\{u \in \mathcal C^\infty (\mathbb R^n) \,|\, supp ...
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17 views

Symmetric Matrix for Shape Operator?

Let $R$ be a smooth surface (smoothly embedded) in $\mathbb{R}^3$. Let $M$ be the matrix for the Shape operator of $R$ with respect to the basis $\{\partial _x F, \partial_yF\}$ for the tangent space ...
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9 views

Metric of the doubling i.e circular billard table

Let $D^2=B_1(0) \cup_{\partial B_1(0)} B_1(0)$ denote the doubling in $\mathbb{R}^2$ i.e. the metric space which one gets after gluing two closed balls of radius $1$ with their induced metric coming ...
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12 views

parameterization of surface of revolution?

parameterization of surface of revolution formed by revolving the $x=\cosh z$ around z axis , i thought the it as $$x=\cosh z \cos \theta ,y=\cosh z \sin \theta ,z=z$$ Hence the surface can be ...
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Showing that a mapping is an isometry

Can anyone please help with this question? I have tried substituting but I'm not sure if that's correct so think I am missing something. Let S denote the surface of revolution $(x, y, z) = ...
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33 views

Surface area of Convex bodies contained in one another

Suppose we have two compact convex bodies one contained in the other in $\mathbb{R}^n$, $C\subset D\subset \mathbb{R}^n$. Does it follow that the ($n-1$ dimensional) surface area of $C$ is less than ...
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22 views

How do I show that the reparametrization of a pre-geodesic is pre-geodesic?

So this is probably a very silly question but I need to prove the reparametrization of a pre-geodesic is a pre-geodesic. The hint in my book says it's obvious, so I am clearly missing something.
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basic question of differential geometry.

I'm taking a course on general relativity and I did a problem that I have now found out I have no idea how to interpret. The surface of a paraboloid has the metric $$ds^2=(1+r^2)dr^2+r^2d\theta^2$$ ...
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26 views

Proper patch in the differential geometry

I have a question that coincides with this question. Proving that every patch in a surface $M$ in $R^3$ is proper. Prove that if $\mathbf{y}:E\to M$ is a proper patch, then $\mathbf{y}$ carries ...
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1answer
30 views

Every compact hypersurface in $\mathbb{R}^n$ is orientable

Show that every compact hypersurface in $\mathbb{R}^n$ is orientable. HINT: Jordan-Brouwer Separation Theorem. This is an exercise from Guillemin and Pollack. So hypersurface means smooth ...
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21 views

Example of a Symbol: Connection.

I'm trying to get more intuition for the symbol of a differential operator. In particular, I've tried the example of a connection. What is the most efficient way to compute the symbol for, say, a ...
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15 views

Finding a zero of homogeneous parts of polynomials using zero of those polynomials

Let $f, g \in \mathbb{Q}[x_1, ..., x_n]$ be polynomials of degree $d$. Let $F$ and $G$ denote the degree $d$ portions of $f$ and $g$ respectively. Suppose there exists a non-singular point ...