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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Groups that are not Lie Groups

What are some examples of groups that can not be given a smooth structure such that they become a Lie Group? Edit: To be a bit more specific, I was hoping that somebody could give an example of a ...
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What is the relation between $C^\infty$-linear and tensorial?

I've been self studying Riemannian Geometry through Spivak's and Lee's books, and fairly often I've seen an argument that goes somewhat like this: We have an operator that acts on vectors in the ...
6
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69 views

Are $T\mathbb{S}_2$ and $\mathbb{S}_2 \times \mathbb{R}^2$ different?

I have seen the claim that $T\mathbb{S}_2$ and $\mathbb{S}_2 \times \mathbb{R}^2$ are not diffeomorphic, but I have only ever seen the proof that they are not isomorphic as vector bundles (which is a ...
2
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33 views

Proof of Hopf's theorem using Liouville

Hopf Theorem A topological sphere immersed as a constant mean curvature surface in $\mathbb{E}^3$ is a round sphere In Heinz Hopf's Differential Geometry in the Large, a proof is given of the ...
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46 views

Normal coordinate parallel along radial geodesics?

A radial geodesic in normal coordinates is given by $\gamma:t \mapsto t(V_1,....,V_n).$ Is it then true that any normal coordinate $\partial_x|_{\gamma}$ is parallel along $\gamma,$ i.e. ...
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Exercise 3.2 of Do Carmo's “Riemannian Geometry”.

I am trying to do exercise 3.2 of Do Carmo's Riemannian Geometry. After constructing the natural metric in the tangent bundle of a Riemannian manifold, he defines "horizontal vector to the fiber". Its ...
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36 views

parallel vector field

I was wondering about the following: I know that a vector field along a geodesic that is parallel has a constant angle to the tangent vector of the curve and constant length. Now, is the converse ...
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2answers
36 views

What does the vertical bar mean in $ \left.\frac{\partial f}{\partial x}\right\rvert $

I want to know what the symbol '|' besides a function means. For example: $$ \left.\frac{\partial f}{\partial x}\right\rvert $$
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59 views

Existence of diffeomorphism through convergence in Hausdorff distance

I'm reading a book and have come across something that I cannot verify or fix. The assumption is that $\Omega_1, \Omega_2, ...$ is a sequence of connected open sets in $\mathbb{R}^n$ that converge in ...
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1answer
36 views

Definition of covariant derivative of a covariant derivative

If we have a connection $\nabla$ different than the Levi-Civita connection, and for a Riemannian metric $g$ and $\nabla$ this relation is valid: \begin{align} ...
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2answers
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How do the components of a cross product transform?

Let $x^{j}$ and $y^{k}$ be the components of two vectors $x,y\in \mathbb{R}^{3}$. According to the way the compontents of $x$ and $y$ transform when we change the basis, we know they are ...
2
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1answer
28 views

Tangent and normal spaces of submanifold of fixed-rank matrices

Let $m \geq 2$. The subset $X$ of $m \times 2$ matrices with rank $1$ is a (smooth) submanifold of $\mathbb{R}^{m\times 2}$. Let $A$ be in $X$. I know from a more general statement that the tangent ...
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35 views

A question about Lie derivative and taking trace

Let $(M,\omega)$ be a complex Kahler manifold, and $g$ is a smooth function such that $\int_Mg\omega^n=0$. It is obvious that there exists a smooth function $f$ such that $\triangle_\omega f=g$. ...
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1answer
32 views

Is it true that all $k$-submanifolds of a $m$-manifold are open subsets of some closed $k$-submanifold?

Let $M$ be a $m$-dimensional (smooth) manifold. I know that $m$-submanifolds of $M$ are exactly the open subsets of $M$. Is it true that all $k$-submanifolds of $M$ are open subsets of some closed ...
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2answers
53 views

a question on topological manifolds and what topology provides

When one talks of a topological manifold being locally homeomorphic to $\mathbb{R}^{n}$ is it meant that the topology of the manifold is locally identical to a Euclidean topology such that we can ...
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2answers
66 views

Is it mathematically correct to say that if the metric is flat/curved the *shortest* path is/not a Euclidean straight line?

Is it mathematically correct to say that if the metric is flat/curved the shortest path is/not a Euclidean straight line? I am still hesitant to make this claim, due to at least one counter example. ...
3
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1answer
139 views

What is a topology?

Having read through the mathematical definition of endowing a set with a topology I must admit that I'm still struggling to conceptualise what such a mathematical construct is. I've read articles ...
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29 views

Decomposition of acceleration into normal and tangential components

If the velocity $v=\|\mathbf{v}\|$ of a point having position $\mathbf{x}(t)$ at time $t$ is never null, then acceleration $\mathbf{a}:=\frac{d^2\mathbf{x}}{dt^2}$ can be written ...
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33 views

Riemannian tensor and Levi Civita connection

For a riemannian metric $g$ consider the following tensor $T_{rstu}=k(x)g_{rt}g_{su}-k(x)g_{st}g_{ru}$. Which condition has to satisfy $k$ if we want the tensor $T$ to be the Riemann tensor of a ...
2
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1answer
29 views

The last coordinates of basis vectors are a chart: mistake in this example?

While trying to understand local chart on Grassmannians I came across this example in this book: Take $V = \mathbb R^2$ and $U,W$ two subspaces generated by linearly independent vectors. The books ...
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1answer
41 views

The last coordinates of basis vectors are a chart

Let $Gr$ denote the Grassmannian and let $Gr(2, T\mathbb R^3) = \bigcup_{x \in T\mathbb R^3} Gr(2, T_x \mathbb R^3)$. Consider one $2$-dimensional subspace of $\mathbb R^3$, that is, one element of ...
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69 views

Surface of constant mean curvature

From PDE Evans, 2nd edition: Chapter 8, Exercise 12: Assume $u$ is a smooth minimizer of the area integral $$I[w]=\int_U (1+|Dw|^2)^{1/2} \, dx,$$ subject to given boundary conditions $w=g$ on ...
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1answer
15 views

Tangent space to noncompact Stiefel manifold

The noncompact Stiefel manifold is the set of $\mathbb{R}^{n \times p}$ matrices ($p \leq n$) that have rank $p$ (full rank). Based on my readings of ...
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49 views

Covariant derivative and box operator commutator

I know that the commutator of two covariant derivatives is giving some Riemann tensors as follow: ...
3
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2answers
27 views

Pushforward injective

Let $f : M \rightarrow N$ be a smooth surjective map between smooth manifolds. Now, consider a 2-form $\omega$ on $T_pN$. Does it now follow that the pullback satisfies? $f^* d \omega =0 \Rightarrow ...
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Deriving a curvature tensor for a special connection

May be $e(x)$ a frame vector at the point $x \in M$ for a manifold $M$. Given the following connection equation: $d_Xe+e_{|X_+}-e_{|X_-}= d_Xe + J_Xe = 0$. Here, $d_X$ denotes the directional ...
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Is there some relationship between algebraic curves and partial differential equations that goes beyond classifying different PDE's

I ask primarily because despite not having taken that many math classes (up to two semesters of a PDE class in college), it would be very interesting if maybe we could gain intuition regarding ...
3
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2answers
59 views

Is the shortest path in flat hyperbolic space straight relative to Euclidean space?

I have the following metric $$ ds^2 = dt^2-dx^2 $$ and I wanted to prove to myself that the shortest path for this metric is straight. I used the following relation $x=f(t)$ and $$ S = ...
3
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2answers
34 views

Exterior derivative of a form and $d(d\omega)=0$?

We know that in differential geometry, $d^2\omega=0$, where $\omega $ is a form and $d$ is the exterior derivative. However if this form happens to be the exterior derivative of another form ...
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1answer
26 views

If $M\subset \mathbb{R}^d$ is a manifold of dimension $m$ and $U\subset \mathbb{R}^d$ is open, then $M\cap U$ is not a open.

I'm reading notes about M-estimators, and have within these notes been briefly introduced to manifolds, as a way to create what the author call "smooth hypothesis" for statistical models. A basic ...
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119 views

Isn't there a better way to put the canonical smooth structure on the $n$-sphere?

My differential geometry notes put a smooth structure on the $n$-sphere (denoted $S_n$) as follows. Firstly, $S_n$ is taken as a subset of $\mathbb{R}^{n+1}.$ Secondly, we define $U_0 = S_n \setminus ...
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2answers
80 views

A “parallel manifold” is always orientable

I want to solve the following problem from Spivak's Calculus on Manifolds: Let $M$ be an $(n-1)$ dimensional manifold in $\mathbb{R}^n$. Let $M(\varepsilon)$ be the set of end points of normal ...
3
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0answers
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Any unit speed reparametrization $\beta=\alpha(h)$ of $\alpha$ is reparametrized by an arc length function.

These are definitions given from Barret O'neill's Elementary Differential Geometry. Definition $1$. A curve in $\mathbb{R}^3$ is a differentiable function $\alpha: I\to \mathbb{R}^3$ from an open ...
2
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1answer
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Question about vectors in the Grassmannian in this example

Consider $f: \mathbb R^2 \to \mathbb R^3$ defined by $(t,s) \mapsto (t^2 + 2s, t^3 + 3ts, t^4 + 4t^2 s)$. Let $Gr$ denote the Grassmannian and let $Gr(2, T\mathbb R^3) = \bigcup_{x \in T\mathbb R^3} ...
3
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1answer
36 views

Mean Curvature Flow

Recently I am reading the mean curvature flow from the lecture notes of Carlo Mantegazza where I found that Under mean curvature flow given by$$\begin{cases}{\partial\over \partial ...
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1answer
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What is the flaw in my thinking for the graph of this function?

Consider the map $$ f: \mathbb R^2 \to \mathbb R, (x,y) \mapsto x^3 + y^3 + xy$$ This defines a surface in $\mathbb R^3$. Let's consider some level set $f(x,y) = c$: (see here page 67) I think ...
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1answer
13 views

Asymmetry of definition of regular value and critical value

Let $f: U \subseteq \mathbb R^n \to \mathbb R^m$ be a smooth map. We say $y \in \mathbb R^m$ is a regular value of $f$ if and only if all points in the set $f^{-1}(y)$ are regular. (see e.g. the ...
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4answers
209 views

How many points can I prescribe for a diffeomorphism of the plane?

I was trying to find out how to construct a $\mathcal C^\infty$ curve that joins two arbitrary line segments. My idea was to use bump functions and the likes, but for that I had to make the line ...
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1answer
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Show that the section $g(x_1,x_2,x_3)=x_1^2dx_1^2+dx_2^2+dx_3^2$ defines a Riemannian metric on $\mathbb{R}^3 - \{x_1=0\}$

Show that the section $g$ of $T^*\mathbb{R}^3 \otimes T^*\mathbb{R}^3$ defined by $g(x_1,x_2,x_3)=x_1^2dx_1^2+dx_2^2+dx_3^2$ defines a Riemannian metric on $\mathbb{R}^3 - \{x_1=0\}$ and compute ...
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31 views

Incorrect statement in a proof of the transversality theorem?

I'm reading through Morris Hirsch's book on differential topology, and he makes the following offhand statement. Suppose k is a compact subset of a manifold U, and V is a vector subspace of R^n. If a ...
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1answer
19 views

Compute $(df)_a$ in chart $\varphi_1:U=\{(x,y,z)\in\mathbb{R}^3:x\neq0\}\rightarrow\varphi_1(U)$

Suppose that for a submanifold $H$ of $\mathbb{R}^3$ we have two charts $$\varphi_1:U=\{(x,y,z)\in\mathbb{R}^3:x\neq0\}\rightarrow\varphi_1(U)$$ ...
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Characterization of Semidirect Products: Too strong assumptions?

In John Lee's book Introduction to Smooth Manifolds, there is the following Theorem. Theorem 7.35 (Characterization of Semidirect Products). Suppose $G$ is a Lie group, and $N,H\subseteq G$ are ...
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1answer
50 views

Properly discontinuous action on manifold

I am actually not familiar with topology, but since we had a short outlook on these things in our differential geometry lecture today, I would appreciate some general remarks: Let $M$ be a smooth ...
2
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1answer
26 views

Multidimensional variant of the fundamental Lemma of the Calculus of Variations

I wonder, if the following is true: Let $(M,g)$ be a compact Riemannian manifold and $f \in \mathcal{C}^{\infty}(M)$ be a smooth function. Then $f$ is constant if and only if for all $u \in ...
3
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1answer
59 views

Coordinate systems on manifolds

I am fairly new to differential geometry and something I can't get my head around is, if an $n$-dimensional manifold is locally homeomorphic to $\mathbb{R}^{n}$, i.e. Euclidean space, then isn't it ...
4
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0answers
71 views

Convex polyhedron and its Gauß-curvature

I am trying to prove: A convex polyhedron has positive Gauß-Curvature at every vertex. What we know: Gauß-Curvature at every vertex is given by $K(p) = 2\pi - \sum\limits_{\text{angle } ...
3
votes
2answers
41 views

Notion of curvature for a volume embedded in $R^3$

This question might sound slightly vague, but please bear with me. If I have an orientable, closed, sufficiently smooth surface in $R^3$, I can define its principal curvatures, mean curvature as ...
2
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1answer
20 views

a curve does not need to be injective?

In diff. Geometry, curve is a differentiable mapping from an open interval to 3 dimensional euclidean space. Doesn't it need to be injective? If it is not, then there might be a two different tangent ...
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1answer
35 views

Find an atlas for $M=\{(x,y,z)\in\mathbb{R}^3:x^2+y^2=1+z^2\}$

Find an atlas for $M=\{(x,y,z)\in\mathbb{R}^3:x^2+y^2=1+z^2\}$ It is easy for me to check that $M$ is a submanifold of $\mathbb{R}^3$ of dimension $2$ using the following theorem: Let $F:U ...
4
votes
1answer
28 views

Calculate the length of $\gamma(t)=(t,t), t \in [-1,-\frac{1}{2}]$ with the metric $g=\frac{dx^2+dy^2}{y^2}$ and compare with euclidean metric

Consider the metric $g=\frac{dx^2+dy^2}{y^2}$ on $\mathbb{R}_+^2=\{(x,y) \in \mathbb{R}^2 : y>0\}$. Calculate the length of the curve $\gamma(t)=(t,t), t \in [-1,-\frac{1}{2}]$ and compare ...