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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About the diameter of a Riemannian manifold

My exact problem is a Riemannian manifold defined on $SU(2^n)$, where the metric is defined as follows: If $U(t)$ is a curve so that $U′(t)=−iH(t)U(t)$ (so just a unitary evolution of a quantum system ...
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69 views

flat manifold, curvature and the circle

A Riemannian manifold is said to be flat if the curvature is 0 everywhere. An example in dimension 1 is the circle. However, I cannot see how the curvature of the circle could be 0. See for instance ...
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Definition of complex submanifold

For smooth manifolds, we can define an embedded submanifold to be either (1) a subset locally cut out by "slice" charts, or (2) a subset that is a manifold in the subspace topology and admits a smooth ...
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38 views

Looking for a biholomorphism [closed]

Is there a biholomorphism between $\mathbb{H}_2 \times S^1$ and a half-space of $\mathbb{R}^3$?
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1answer
30 views

Diameter of a Riemannian manifold on $SU(N)$ with almost negative curvature everywhere.

Are there any results (papers/books) on this problem? I am working on a finite dimensional Riemannian manifold which has a negative curvature almost everywhere. But I do not know if such kind of ...
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34 views

Why is the singular $p$-simplex called singular?

I am learning homology. I cannot see in which sense the singular $p$-simplex is singular. It seems quite regular. Any idea?
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38 views

Reference request: Pullbacks along submersions are submanifolds and the induced map is a submersion

I'm looking for an introductory book in differential topology in which there are proofs of the following facts: Let $X,X',Y$ be smooth manifolds, $X\rightarrow Y$ a smooth map and $X'\rightarrow Y$ ...
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1answer
35 views

Vector field on $S^2 \setminus \mathsf{NP}$ looks like a magnetic dipole

The following is a question from Spivak's Differential Geometry text: Not really sure what he's going for here. Any ideas?
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1answer
37 views

The Differential Geometry of a 2-D Surface

I'm currently self-studying the differential geometry of embedded surfaces. My question is, how am I to chose the appropriate coordinates and derive the covariant basis for the surface I'm interested ...
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34 views

Relation between integral curves

Let $M$ be a manifold, $f:M\rightarrow\mathbb{R}$ a differentiable map and $X$ a vector field on $M$. I'm trying to find a relation between the integral curves of $X$ and $e^fX$. I am not quite sure ...
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44 views

What if there is $\downarrow$ or $\uparrow$ notation in the limit instead of $\rightarrow$?

I saw a different notation in a limit in the book Elementary Differential Geometry by A N Pressley : what do both of $\downarrow$ and $\uparrow$ mean?
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1answer
26 views

How does an atlas give a notion of whether a function is differentiable or not?

Defintion. An atlas is a set of pairs $\{(O_i,ψ_i)\}$ such that $\cup_iO_i=M$ and $\phi:O_i \rightarrow \mathbb{R}^n$, and most importantly, for $O_i \cap O_j \neq \emptyset $, $\tau_{ij}: \phi_i(O_i ...
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I have no idea what “smooth structure” is

I know what a manifold is: it's a topological space such that for every point there is an open set that looks like $\mathbb{R}^n$. But I do not know what a smooth manifold is, because I have no clue ...
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Volume forms and volume of a smooth manifold

Choose a volume form $\omega$ on $M$, oriented manifold. For every $F\in C^{\infty}_c(M)$, we define $$ \int_M F:=\int_M F\omega $$ where in the right hand term $M$ is taken wit positive orientation ...
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1answer
88 views

Manifold is not orientable

Let $M$ be a manifold of dimension $n$ such that there exist two charts $(U_a,\phi_a)$ and $(U_b,\phi_b)$ such that $U_a,U_b$ are connected and $U_a\cap U_b\ne\emptyset$. Moreover the ...
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1answer
31 views

vector field on $\mathbb{R}^n$ versus on manifold

I am looking for a counter example that why the $\mathbb{R}^n$ definition of vector field fail on a manifold. The following is a summary of what I learnt few years ago. Start with the idea of ...
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24 views

Representing a vector field locally

A couple of questions occured to me when reading about the Poincare-Hopf theorem. Any pointers or comments would be appreciated! Let $M$ be a closed oriented Riemannian manifold and $V$ a vector ...
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Curve as a linear combination of Frenet frame variables

How to write a curve $\alpha(s)$ as a linear combination of $\alpha'(s),n(s),b(s)$ where these are the tangent to the curve, the normal vector and the binormal vector. Where torsion is nonzero and ...
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1answer
38 views

Equivalent conditions for a linear connection $\nabla$ to be compatible with Riemannian metric $g$

I am reading John M. Lee's Riemannian Manifolds: An Introduction to Curvature. In Lemma $5.2$, it is said that the following conditions are equivalent for a linear connection $\nabla$ on a Riemannian ...
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40 views

Integral curves on non compact manifolds

Define a vector field on $\mathbb{R}^d$ by $X = \frac{\partial}{\partial x_{d}}$. That is a vector field that always points upward along the $x_{d}$-axis. Consider starting at any point $p \in ...
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Gluing two diffeomorphisms

Let $M_1,M_2$ be two $n$-dimensional closed manifolds and suppose that $M_i=\bar{U}_i^+\cup \bar{U}_i^-$ where $\bar{U}_i^\pm$ are compact codimension zero sub-manifolds with boundary so that ...
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1answer
24 views

The definition of a functional structure on a topological space

I am reading Glen E. Bredon's Topology and Geometry and trying to solve the following problem in the book. But I can't understand what the blue g means. Could anyone explain please? Show that a ...
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1answer
131 views

Parallelizability of Lie groups

I have tried a lot to prove this well-known result. The basic idea behind the proof is clear to me. But I'm stuck at showing either of the following conditions: The map $\ G \rightarrow TG \ $ given ...
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2answers
65 views

Can $S^4$ be the cotangent bundle of a manifold?

I am asking the question because in the classical mechanics book by Arnold, he states that there is a distinguished $1$-form on $T^*V$. It seems that there is no such distinguished $1$-form on a ...
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1answer
23 views

Can $S^4$ be the cotangent bundle of a manifold?

I am asking the question because in the classical mechanics book by Arnold, he states that there is a distinguished 1-form on $T^*V $. It seems that there is no such distinguished 1-form on a general ...
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27 views

Lie group action from the Lie algebra

want to find the corresponding lifting f the standard U(n) lifting on $C^n$ to $L=C^n \times C$ with hermitian metric $e^(-|z|^2)$. I try to follow the method in Donaldson, and I find if B in u(n) ...
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Proof a special extension on the isoperimetric theorem

Let both ends of a string of length $L$ be tied to a stick of length $S$. Among all plane regions enclosed by this contraption, it achieves maximum when the string forms a circular arc. It is noted ...
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41 views

Preserving arc length for a family of curves

Sorry if I have formatted things wrong, I have read the tour and browsed around, so I tried my best. I have a one parameter family of curves with the relation: $$\frac{\partial}{\partial \lambda} ...
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2answers
51 views

Do two embeddings of a Euclidean space into a higher dimensional one only differ by a diffeomorphism?

Let $d\le n$ and $$f,g\colon\mathbb{R}^d\hookrightarrow\mathbb{R}^n$$ be two smooth embeddings. Is there a diffeomorphism $$\phi\colon\mathbb{R}^n\rightarrow \mathbb{R}^n,$$ such that $$f=\phi\circ ...
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33 views

continuous (smooth) maps and group homomorphism

Consider a topological group $G$ (or smooth Lie group) and a topological space $M$ (or smooth manifold) and a group homomorphism $\phi:G\rightarrow Sym(M)$, where $Sym(M)$ is the symmetry group of M, ...
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1answer
43 views

Finding the curvature and normal vector for an arclength curve

I am trying to find the curvature and normal vector for $$\alpha(t) = \left(\frac13 (1+t)^{3/2}, \frac13 (1-t)^{3/2}, \frac{t}{\sqrt2}\right)$$ $$\alpha'(t) = \left(\frac12 (1+t)^\frac12 , - \frac 12 ...
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Curvature and Torsion explained

Let $\alpha: I \to \Bbb R^3$ be a curve parameterized by arc length $s$, with curvature $k(s)\ne 0$, for all $s\in I$ $$\alpha(s) = \left(a\cos \frac sc, a \sin \frac sc, b\frac sc\right),\quad s\in ...
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Formal adjoint of curvature (Yang Mills)

Currently reading a paper on finding solutions to the Yang Mills equation $D^*\Omega=0$, where $\Omega$ is the curvature and $D^*$ is the formal adjoint of the exterior covariant derivative $D$. ...
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34 views

Differential Geometry concept verification

If I have a regular parameterized curve: $$\alpha(t)$$ The curvature, $k(t)$, is precisely $\|\alpha''(t)\|$, The normal vector, $n(t)$ is found by looking at $\alpha''(t) = n(t)k(t)$ The binormal ...
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Prove the theorem on analytic geometry in the picture.

I discovered this elegant theorem in my facebook feed. Does anyone have any idea how to prove? Formulations of this theorem can be found in the answers and the comments. You are welcome to join in ...
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1answer
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Typo in “Intro to Contact Topology” by Geiges, Lemma 1.4.10?

In Introduction to Contact Topology by Geiges, there is a result relating Hamiltonian and Reeb flows for hypersurfaces of contact type in a symplectic manifold. Lemma 1.4.10 $\,$If a codimension 1 ...
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Singularity in Ricci flow vs Ricci soliton

In the paper "The formation of singularity in Ricci flow" Hamilton studied systematically the possible singularities of the flow.My question is why it is important to classify Ricci solitons in order ...
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50 views

$V$-bundles and vector bundles

I am looking for more information on $V$-bundles. They are hard to search for as either vector bundles come up or something like GL($V$)-bundles come up. I am looking for some nice expository ...
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Solving a 2nd-order elliptic PDE with non-constant coefficients

I wonder how I can solve the following 2nd-order PDE on the positive semiplane $\{x>0\}$: $$(\partial_x^2+\frac{1}{x}\partial_y^2)\phi=\delta(x-x_0)\delta(y).$$ I notice that the l.h.s. is the ...
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40 views

Level sets on $SU(n)$

Given $G \in SU(n)$, what are the level sets of the function $F(V) = |tr(G^{\dagger}V)|^2$? Can they be written only in terms of abstract linear maps, not in terms of the components of $V$ and $G$ in ...
2
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1answer
54 views

Scalar product on Lie algebra of compact Lie group [duplicate]

I am studying Differential Geometry and I am facing with a lemma in which there is a step that I do not understand. In particular, let $G$ be a connected compact Lie group, is used "$\langle\ \cdot , ...
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1answer
39 views

Is it meaningful to take “exterior products” of vector fields?

Let $M$ denote a smooth manifold. I've read that a differential $k$-form is a smooth section of the $k$th exterior power of the cotangent bundle of $M$. However I barely understand what this means, ...
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1answer
38 views

Alternative Proof of why Every Manifold is Locally Compact

So while I was solving some problems on differential geometry, I stumbled upon a problem which is to show that every manifold is locally compact. Now, there is a proof for it here, but I was thinking ...
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30 views

A detail in the proof orientable manifold admits exactly two orientations

Let $M$ be a real manifold and let $\{(U_\alpha,\phi_{\alpha})\}_{\alpha\in I}$ $\{(V_\beta,\psi_\beta)\}_{\beta\in J}$ be two oriented atlases. Let's define, for $p\in U_\alpha\cap V_\beta$, ...
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Parallel transport along a 2-sphere.

I'm currently learning about parallel transport and connections and we were considering the parallel transport of a tangent vector along a sphere as given in the picture below. From my ...
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1answer
33 views

“Winding number”, Chern character and relative signatures of the metric

Anyone answer with good explanation is appreciated. In differential geometry, we discuss about topological quantities like characteristic classes. For example, the first Chern character of some ...
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Locus of points on a rotating line ; points differently ordered

A line rotates about a fixed point $O$ with ordered points $P,O,M $, while $ M $ is moving along this line $POM$. Find locus of points $ P ,M $ if $ MP^2- OM^2 = T^2 $ constant for all inclinations ...
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1answer
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Local extension of a function on an immersed submanifold

Consider the following passage in Spivak's Differential Geometry book: I am having trouble understanding where he says $g = \tilde{g} \circ i$ on $V \cap M_1$. Since $V$ is (I think) supposed to be ...
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34 views

Show that $1$-form has particular coordinate representation.

I want to show that for a nowhere-vanishing $1$-form $\alpha$ on an $n$-dimensional manifold $M$ we have that $\alpha \wedge d\alpha = 0$ if and only if around every point $p$ there exists a ...
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269 views

Can the interior of a manifold be orientable but not its boundary?

Suppose $M^m$ is a manifold with boundary. If we are given an orientation for $M$, we can then derive an orientation for $\partial M$ by considering the orientation of $TM$ at $\partial M$ and then ...