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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Defining a Riemannian metric

Let $M$ be a smooth manifold. I have seen a Riemannian metric be defined in many ways: A smooth choice of an inner product $g_p:T_pM\times T_pM\to\mathbb{R}$ which is symmetric and positive-definite,...
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27 views

Clarification for Manifold and Manifold with Boundary Definitions

I am reading Spivak's Calculus on Manifolds, and just need a bit of clarification when it comes to definitions. He defines a manifold as some space $M$ that satisfies: $(M)$: $\forall x \in M, \...
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24 views

Need help understanding part of this proof about local coordinates for Legendrian manifold

I need help understanding this proof in this book here: Concretely, I do not understand why it is okay to assume that $S$ can be parameterized by $n$ variables. Sure, it's an $n$-dimensional ...
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34 views

Equation of silhouette from an arbitrary viewpoint

A two parameter $(u,v)$ surface in $\mathbb R^3$ when viewed from a point at infinite distance casts a shadow on any given plane. What ODE/PDE describes its envelope of its silhouetted projection? ...
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36 views

Why is the Christoffel symbol of the 2nd kind symmetric in lower indices?

I have consulted multiple books on tensors for physicists, but they all take for granted this relation: $\Gamma_{ij}^k = \Gamma_{ji}^k$ However, no proof is provided and I cannot find a single one ...
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1answer
19 views

Given parametrization of torus is equal to level surface

I need to show that the torus $T=\pmatrix{(R+r\cos\phi)\cos \theta\\(R+r\cos\phi)\sin\theta\\r\sin\phi}$ is equal to the surface given implicitly by $(\sqrt{x^2+y^2}-R)^2+z^2-r^2=0$. I already got ...
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1answer
63 views

Does the associated bundle functor have left or right adjoints?

Let $\mathsf{Prin}_G$ be the category of (right) $G$-principal bundles, with a morphism from the bundle $p: P \to M$ to the bundle $p': P' \to M'$ being a pair of arrows $\chi: P \to P'$ and $\bar{\...
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45 views

Reference about the surgery of Ricci flow

I roughly read the Topping's LECTURES ON THE RICCI FLOW. Seemly, there is not introduction about surgery. Seemly,it is enough to deal singularity by blow up. Then, for knowing surgery I read the ...
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39 views

Pullback Courant algebroid

I am reading the lecture notes on generalized geometry http://www.staff.science.uu.nl/~caval101/homepage/Research_files/australia.pdf but now I am stuck on the exercise 1.31 at page 13. Here is the ...
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1answer
30 views

Compute Christoffel symbols of a rotating cartesian coordinate system

Suppose we have a smooth manifold $(\mathbb{R}^3, \mathcal{O}_{\mathbb{R}^4}, \{(\mathbb{R}^3,x),(\mathbb{R}^3,y)\},\nabla,t)$ where $t:\mathbb{R}^3\rightarrow\mathbb{R}$ is such that $t(a,b,c,d)=a$, $...
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29 views

Smooth structure in reconstruction theorem

Let $M,F$ be smooth manifolds, $\{U_i:i\in I\}$ an open cover of $M$ and a cocycle $\{t_{ij}:U_i\cap U_j\to\mathrm{Diff}(F)\}$. In almost any book which discusses fibre bundles, one can find the ...
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1answer
26 views

Curve in a disk

I take a curve $\vec\gamma:[a,b]\longrightarrow \Delta$, where $\Delta \subseteq \mathbb{R}^2$ is the disk of radius $r>0$. If the curve has length $L>0$ does exist an upper bound (in terms of $...
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1answer
30 views

Do these assumptions on a mapping ensure it is a diffeomorphism?

$A\subset \mathbb{R}^m$ is an open and bounded set, $f:A\longrightarrow \mathbb{R}^n$, $m\leq n$, is injective, continuously differentiable and its Jacobian matrix has full rank on $A$. Does this ...
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1answer
49 views

Is a surjective mapping of R2 to itself with full rank derivative everywhere necessarily injective?

If $f:\mathbb R^2\rightarrow\mathbb R^2$ has rank 2 derivative everywhere, then by the inverse function theorem it is locally injective. If it is surjective, is it then necessarily globally injective ...
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17 views

Unique Solution to Equation in Two Variables & Possible Use of the Implicit Function Theorem

Let $g(x) : R \to R$ be a continuous function; Consider the equation $ T(x,y) = y^3 -y^2 +(1+x^2)y - g(x)$ Show that for a given $x$ there exist a unique solution $y$ to the equation $T(x,y) = 0$. ...
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1answer
46 views

Definition of formal adjoint of covariant derivative

I read in Einstein Manifolds, L. Besse that the covariant derivative $D: \mathcal{J}^{(r,s)}(M)\to \Omega^1(M)\otimes\mathcal{J}^{(r,s)}(M)$ admit an formal adjoint $D^*:\Omega^1(M)\otimes\mathcal{J}^{...
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29 views

curl-free vector field on 3-torus

Let $U$ be an open and simply-connected subset of $ \mathbb{R}^3$. Then for every curl-free vector field $v \: \colon U \to \mathbb{R}^3$ there is a potential $\phi \in C^{\infty}(U; \mathbb{R})$ such ...
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1answer
29 views

Geodesic sphere in $\mathbb H^2$

I saw the definition of a geodesic sphere, and I think I'm not able to "see" how do they look like. For example, it's obvious that in $\mathbb R^n$ geodesic spheres are simply normal spheres, and that ...
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1answer
14 views

Existence of a $G$-invariant metric on a principal bundle

Given a smooth principal $G$-bundle $\pi: P \rightarrow M$, I want to show the existence of connections by showing that $P$ admits $G$-invariant metrics. I was thinking in a kind of averaging ...
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30 views

How to prove that a space is not a differential manifold?

Given a box (the surface of a cubic) in R^3 space, can I give a smooth structure on it to make it a differential manifold?
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158 views

Reference of what metric can be placed on manifold?

I just read some conclusion that $T^2$ can't be placed metric with positive curvature at all points. I don't know why is so . And what book introduce about this ? I mean about what metric can be ...
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54 views

Is this what universal covering spaces are used for?

From the perspective of real analysis, we have: $$\int_{-1}^{1}\frac{1}{1+x^2} = \mathrm{tan}^{-1}(1)-\mathrm{tan}^{-1}(-1) = 2\mathrm{tan}^{-1}(1) = 2 \cdot \frac{\pi}{4} = \frac{\pi}{2}$$ Something ...
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40 views

Deviation of a plane curve from the $x$-axis

I have a smooth plane curve $\gamma$ enclosed in a circle of radius $R$. See Figure 1. I'm interested in measuring the deviation of this curve from the $x$ axis, denoted $\Delta(t)$ in the figure, as ...
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210 views

Which concepts in Differential Geometry can NOT be represented using Geometric Algebra?

1. It is not clear to me that linear duals, and not just Hodge duals, can be represented in geometric algebra at all. See, for example, here. Can linear duals (i.e. linear functionals) be ...
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10 views

local sections transformation formula

I am trying to prove a formula which states the relations between two local sections in a principal bundle: Let $P(M,G)$ be a principal bundle let $\{ U_\alpha \}_{\alpha \in I}$ be an open cover for ...
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28 views

Volume of a geodesic ball in a Riemannian manifold with $K<0$.

Let M be a simple connected Riemannian Manifold wih $K_M < 0$. Prove that the volume of any geodesic ball of M is strictly greater than $\frac{Vol(S^{n-1})r^n}{n}$, where $n = dim(M)$ and $r$ is ...
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1answer
43 views

Diffeomorphism onto a $k$-manifold in $\mathbb{R}^n$

If $A\subset\mathbb{R}^k$ and $B\subset\mathbb{R}^n$, with $k\leq n$, and $A$ is an open set, then for $f:A\longrightarrow B$ to be a diffeomorphism it must be bijective, continuously differentiable ...
3
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1answer
51 views

How to find the transition function for two overlapping charts of $\mathbb{R}P^2$?

The real 2-dim projective space $\mathbb{R}P^2$ can be covered by the following 3 sets of unoriented lines through the origin un $\mathbb{R}^3$: $ U_x \doteq $ { all lines not lying in the yz plane} ...
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12 views

First and Second Fundamental Form - Direction of Parametrisation

I just have a small doubt. Do the First and Second Fundamental forms depend on the direction of parametrisation? I know that when completely different parametrisations are used, the fundamental ...
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1answer
84 views

Difference between a manifold and a sub-manifold of the same dimension?

I appologize in advance in case this is a very trivial issue and for any mistakes due to translating stuff from my German lecture notes to English ... A subset $M \subset \mathbb{R}^n$ is defined to ...
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1answer
48 views

Partial Derivative of Line Integral as a Potential of F

Context to the question: Say $ \{F_{k} \} \to F$ uniformly on a compact subset $K \subset T$, for $ \{F_{k} \}$ a sequence of conservative vector fields and $T$ open and connected. I've shown that ...
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1answer
45 views

Parallelizable open dense subset and integration

In Petersen's Riemannian Geometry (2016), it is stated on page 8 that any manifold $M^n$ has an open dense subset $O$ with $TO=O\times\Bbb R^n$. Thus it is orientable and one may define the integral ...
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1answer
58 views

Tensor fields on a manifold

Let $M$ be an $n$-dimensional smooth manifold. It is easily shown that the modules $\Gamma(TM)$ (the real vector space of vector fields on $M$) and $\Gamma(T^\ast M)$ (the real vector space of $1$-...
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1answer
30 views

Parallel transport

Suppose $M$ is a manifold with a connection $\nabla$. Fix $p\in M, v\in T_p M$ and let $B$ be a small neighborhood of $p$ such that every $x\in B$ can be joigned with $p$ by a unique geodesic $\gamma_{...
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How can I compute a presentation of the tangent bundle for a smooth manifold defined by a family of polynomials?

Consider a smooth manifold $M$ given by a system of smooth functions $$ \begin{align*} f_1 = 0 \\ \cdots \\ f_k = 0 \end{align*} $$ in $n$ variables. This has the algebraic description as the $\mathbb{...
2
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1answer
35 views

Reductive homogeneous spaces

If $G$ is a connected Lie group and $K$ is a closed subgroup of $G$ then $G/K$ is a homogeneous space. If $\frak g,k$ are the lie subalgebras of $G,K$ resp. Then under the projection $\pi:G\rightarrow ...
3
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1answer
49 views

How is “expressing” a differential operator “in cylindrical coordinates” rigorously defined?

I'm a mathematician (with little knowledge of differential geometry) trying to study physics. One of the greatest problems is the language regarding coordinate transformations. I tend to think of such ...
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0answers
33 views

tangent space in a moving coordinate frame

I've got a problem in some geometry of flow. For the sake of completeness I will give the complete derivation of the equation of interest, but I will seperate it into derivation part and question ...
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1answer
62 views

Cartan decomposition diffeomorphism at the level of (compact) groups

I am trying to understand the Cartan decomposition in the case of a compact group $G$ with subgroup $K$, such that their respective Lie algebras $(\mathfrak{g},\mathfrak{k})$ correspond to a Cartan ...
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1answer
67 views

How to build smooth variations along arbitrary paths at the endpoint?

Let $M$ be a smooth manifold, $\gamma:[0,a] \to M$ a smooth path. Assume we are given another smooth path $\phi:(-\epsilon,\epsilon) \to M$ that starts from an endpoint of $\gamma$, i.e $\phi(0)=\...
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38 views

Proof of iff condition of harmonic map

How to compute the equation above the red line in the picture below ? Below picture is from the Harmonic maps and their heat flows. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~...
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1answer
39 views

There is no smooth retraction from an oriented compact manifold to its boundary

That there is no $C^1$ retraction from a compact, oriented manifold to its boundary is a common lemma in proving a weaker version of the Brouwer fixed point theorem. I recall seeing in class a simple ...
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1answer
30 views

Need a very simple example of coordinate functions and parameterization of a manifold

This is a very simple question from introductory differential geometry. Suppose I have an 2-dimensional manifold $M^2$ that is, for simplicity, a subset of $\mathbb{R}^2$. Now suppose $(U,\phi)$ is a ...
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1answer
32 views

The projective space as a homogeneous space

I want to understand why the projective space $\mathbb RP^n$ is diffeomorophic to $SO(n+1)/O(n)$? and why we can write the latter as $O(n+1)/O(n)\times O(1)$?
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1answer
29 views

Convex combination of projection operators

If $P_1, P_2: V \to V$ are linear projection operators on the vector space $V$ with $R := P_1(V) = P_2(V)$, is it true that any convex combination of $P_1$ and $P_2$ is again a projection operator $...
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1answer
36 views

Is integration with respect to spherical measure equivalent to manifold integration over sphere?

Let $S$ be an $n$--sphere $\mathcal{S}^n(0,R)\subset\mathbb{R}^{n+1}$, $\Theta\subset\mathbb{R}^n$ an open subset and let $\phi:\Theta\subset\mathbb{R}^n\longrightarrow S\subset\mathbb{R}^{n+1}$ be a ...
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20 views

Planar intersections of constant Gauss curvature K surfaces

Have they been studied? It appears they did not generate enough interest except the conic sections $K=0$. Do they give rise to curves of fourth order?
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1answer
26 views

About transformations of the metric: should we use the old or the new one to raise/lower indices?

Let $(M,g)$ be a (Pseudo-)Riemannian manifold. If I perform a transformation on the metric, getting a new metric $\tilde{g}$, which metric should I use to raise and lower indices? As I understand, ...
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1answer
43 views

Confused About Indices in Deriving Curvature

Asking about a step regarding indices in deriving the Curvature tensor from the geodesic equation. Starting from $$ \frac{d v^a}{du} = - \Gamma^a_{bc}v^b \frac{dx^c}{du}$$ we integrate $$v^a(u) = ...
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21 views

I want to show that $ \mid D pr(m) \mid \leq \dfrac{1}{1-\Vert pr(m)-m \Vert \Vert h_{pr(m)}\Vert}.$

Let M be a smooth surface, and $U$ a neighborhood where the orthogonal projection pr is well defined. I would like to show that $ \forall m \in U$, if $ m \in S$, pr is differentiable on m and we ...