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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272 views

How to decide whether F is orientation-preserving or orientation-reversing as a diffeomorphism onto its image.

Let $U$ be the open set $(0,∞)×(0,2π)$ in the $(r,θ)$ -plane $R^2$. We define $F : U ⊂ R^2 → R^2$ by $F (r, θ ) = (r cos θ , r sin θ )$. How to decide whether F is orientation-preserving or ...
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182 views

Extension of the Sylvester-Gallai theorem?

The Sylvester-Gallai theorem asserts that given a finite number of points in the euclidean plane, either: the points are collinear there exists an ordinary line (i.e. a line that contains exactly ...
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335 views

When does a vector field admit orthogonal fields?

My question is: Let $\,X$ be a nonvanishing smooth vector field over an open subset $U \subset \mathbb{R}^3$. Which conditions on $X$ guarantee the existence of a smooth nonvanishing vector field ...
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46 views

How do I show that $F^{∗}(dx∧dy∧dz) = ρ^{2} \sin φ dρ∧dφ∧dθ$.

I dont know how to solve. Please help me. I need to understand such types of the question for my exam studyings.
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118 views

Conditions for a projection of a Knot to be a Knot diagram.

friends. I'm working on a problem, the broad scope of which is to show that given a map $f:S^1\rightarrow \mathbb{R}^3$ be a smooth embedding, and a projection map $\pi_v:S^2\rightarrow P_v$, where ...
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1answer
79 views

Willmore energy of an ellipsoid

Given an ellipsoid of equation $$\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$$ How can I calculate the Willmore energy of this surface knowing that its definition is: ...
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97 views

$F(x,y) = (x^2 +y^2,xy)$. compute $F^{∗}(u \, du+v \, dv)$

Let $F : \Bbb R^2 → \Bbb R^2$ be given by If $u$,$v$ are the standard coordinates on the target $\Bbb R^2$, compute $F^{∗}(u \, du+v \, dv)$. $$F(x,y) = (x^2 +y^2,xy).$$ I am confused so much. I ...
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77 views

Pole of differential

Let $E : y^2 = x^3 + ax + b$ be an elliptic curve over the field $K$, char $K \ne 0$. We know that the differential $\omega = \frac{dx}{y}$ is holomorphic in infinity because we can write it as ...
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83 views

A question about Moebius strip

The Moebius strip (without boundary) $ S $ can be realized as a regular surface of $ R^3 $ (regular surface is meant in the sense of do Carmo's book). Using a 'manifold' language it can be proved that ...
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189 views

Finding the Riemann curvature tensor of the induced metric

The full problem is: Let $(x,y,z)$ be Cartesian coordinates in $\mathbb{R}^3$. Let $x,y,z$ all be a parameterization of a surface $M$ in local coordinates $(u,v)$. Let local coordinates $(u,v)$ be ...
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3k views

Maximum sum of angles in triangle in sphere

Recently my differential geometry lecturer demonstrated that the sum of the interior angles of a triangle in a sphere is not necessarily never $180^\circ$. This is one way to prove that the earth is ...
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122 views

When is a topological space a manifold?

I'm looking for someone to point me in the direction of papers or books which discuss when a topological space (perhaps with the conditions locally compact, Hausdorff) is a topological manifold, ...
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115 views

Deformation retract

How to prove that $r_t$ is a deformation retract $M^a=\lbrace q\in M ; f(q)\leq a\rbrace$ We have the definition : $r_t$ is a difformation retract if: $r_t$ is a continius ,onto application ...
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1answer
450 views

Show that this is a diffeomorphism

I have a function $F:(0,2\pi) \times \mathbb{R}_{>0} \rightarrow \mathbb{R}_{>0}^2$ with $(\phi,r)\mapsto(r(\phi-\sin(\phi)),r(1-cos(\phi)))$ and want to show that this is a smooth(meaning ...
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301 views

Constant Rank theorem for domain with nonempty boundary

Problem 4-3 in J.M. Lee's introductory text about smooth manifolds, asks to formulate and prove a version of the constant rank theorem for a map of constant rank whose domain is a smooth manifold with ...
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56 views

How to estimate error pattern of a set of line segments with respect to given reference segments (2D case)

I am having set of pair of line segments (2D). Though each pair should be coincided on top of each other they are not so. I have been the reference data and then I extracted other line segments ...
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117 views

Nondegenerate critical point

I don't understand this part from the book of Zeidler , can someone help me to understand it ? Please Thank you
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71 views

Curvature form projective spaces

Let $T\mathbb{C}P^n$ tangent bundle over $\mathbb{C}P^n$. We have an hermitian metric on $T \mathbb{C}P^n$ defined as $h=\frac{dzd\bar{z}}{(1+|z|^2)^2}$. If we consider Levi-Civita connection we can ...
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217 views

Relation between triangle and its circumscribed circle, on the surface of a sphere. Generalizations to higher dimensions.

Consider the unit sphere in $R^3$ and an equilateral triangle of side length 1, with all vertices on the surface of the sphere. Now project (from the center of the sphere outward) the triangle and its ...
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56 views

How can I align the angle between points with the magnetic heading as the points move?

I have 3 robots which must track a point. The distance between all the robots and the point is known so a triangle can be formed between any 2 robots and the point. If I find the angles in the ...
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39 views

prove that $supp(π^{∗} f) = (supp f)×N.$ Please can you check my answer? Also more explanation please.

My question is that Let $f \colon M \to R$ be a $C^{\infty}$ function on a manifold $M$. If $N$ is another manifold and $π \colon M \times N \to M$ is the projection onto the first factor, prove ...
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94 views

How are the isometries $h:(\mathbb{R}^n,||\cdot||_p)\longrightarrow(\mathbb{R}^n,||\cdot||_p)\;$?

An isometry of $\mathbb{R}^n$ is a function $h:\mathbb{R}^n\longrightarrow\mathbb{R}^n$ that preserves the distance between vectors: $$||h(x)-h(y)||_p=||x-y||_p\;\;, \;\;p\ge1$$ for all $x$ and $y$ ...
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222 views

How to show that this set isn't a regular surface?

I'm trying to solve this exercise from Do Carmo's Differential Geometry of Curves and Surfaces, and I want a hint on how to do it. The exercise is: Is the set $S =\left\{(x,y,z)\in \mathbb{R}^3 \mid ...
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144 views

Non unique solution for Ricci flow equation

Why completeness is important for the uniqueness of solution to Ricci flow? For example, if $M$ is the open unit disk in $\mathbb{R}^2$ and $g(0)$ is the Euclidean metric, and hence not complete. Why ...
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376 views

Reversing the Ricci flow

Suppose $S$ is a closed, oriented surface (2-manifold) embedded in $\mathbb{R}^3$, which inherits the metric from $\mathbb{R}^3$, so that distances are measured by shortest paths on the surface. If ...
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1answer
180 views

evolution of curvature under ricci flow , What does the tensor A*B means?

in the book of bennet chow,the definition of the "*" notation is simple ,I can't understand . can someone gives me some examples of this definition? or tell me in which book I can find this?thanks.
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130 views

Visualize soliton solutions of a PDE

In trying to visualize soliton solutions of a PDE I faced this sentence: We now think of solitons as self-similar solutions, i.e., solutions which evolve along symmetries of the flow. Question 1: ...
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104 views

Geodesics and Christoffel symbols

If these are satisfied then we are on a geodesic. Do I just need to plug in the condition given about the christoffel symbols and then see that the equations are allways fulfiled as long as $v=at+b$ ...
2
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1answer
283 views

Incomplete vector field

Is there a way I can tell if a vector field on a manifold or just $\mathbb{R}^n$ is incomplete simply by just looking at its formula. For instance on $\mathbb{R}$, $\displaystyle X= (x^2+1) ...
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33 views

Equivalence class involving Lie Brackets..

Can anyone help me with a proof, I spent hours on it and nothing: I want to show $$\overline{[fX, Y]}=f\overline{[X, Y]},$$ where $X$ and $Y$ are smooth vector fields on a smooth manifold $M$, $f\in ...
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2answers
132 views

Extending a smooth map

When can I extend a smooth map $f:\mathbb{R^2}-\lbrace 0 \rbrace \to S^1$ to a smooth map $\tilde{f}:\mathbb{R^2} \to S^1$. For instance, consider $g(x,y)=(x,y)/\sqrt{x^2+y^2}$? Am I able to extend ...
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1answer
70 views

How to make a $C^1$ knot into a $C^\infty$ knot

Suppose I have a $C^1$ imbedding $f: S^1 \rightarrow S^3$. From the point of view of knot theory, what's the "best" way to get a $C^\infty$ curve that "looks like" or is "equivalent to" $f$? For ...
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1answer
68 views

Many partitions of unity on sufficiently “nice”; what does this mean?

In a class I am taking, we are told that are manifolds have "many" partitions of unity (we assume paracompact, Hausdorff, second-countable). However, the course content is not related to this subject. ...
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234 views

Chern classes tangent bundle.

I studied Chern classes but I don't find explicit examples of calculation. So I'd like to consider the tangent bundle over $\mathbb{C}P^n$ ($T\mathbb{C}P^n$) and develop the theory beginning from this ...
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1answer
43 views

“WLOG” when studying Schwarzschild geodesics

Why, when studying geodesics in the Schwarzschild metric, one can WLOG set $$\theta=\frac{\pi}{2}$$? I assume it is so because when digging around the internet, most references seem to consider this ...
2
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1answer
156 views

Möbius maps and their fixed points

Why is it true that if a Möbius map, $f(z)$ fixes distinct $z_1,z_2\in \mathbb C_\infty$ that $f(z)$ either describes a rotation or has a pair of stable and unstable fixed points, such that iterations ...
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1answer
139 views

Is Whitney sum of vector bundle a categorical colimit?

We known that the direct sum of two vector spaces is the categorical colimit of vector spaces. My question is whether Whitney sum of vector bundle is a categorical colimit (in the category of vector ...
8
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1answer
194 views

Triangulation of a manifold adapted to a submanifold

I am not extremely proficient in topology, and am concerned with the following question : Given a compact manifold $X$ and a submanifold $Y \subset X$, is it always possible to find a triangulation ...
6
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1answer
480 views

Cigar soliton solution

In wikipedia, it has been written that an important 2-dimensional example of Ricci flow over $M=\mathbb{R}^2$ is given by $g((x,y),t)=\frac{dx^2+dy^2}{e^{4t}+x^2+y^2} \;\;\; (\star) $ Here are my ...
2
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1answer
81 views

Help with algebraic manipulation to prove that $\Omega (M)$ constructed from $\Omega^1 (M)$ forms an algebra over $C^\infty (M)$

In Baez´s Gauge Theories, Knots and Gravity he states that the differential forms on a n dimensional manifold M, $\Omega (M) = {\bigoplus}_p \Omega^p (M)$, constructed from $\Omega^1(M)$ and the ...
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1answer
136 views

Basic question about definition of Chern classes

Apologies if there is something I missed with a quick internet search, but why do we define Chern classes for complex vector bundles (instead of real vector bundles for example)? If we define chern ...
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128 views

Curvature (Gaussian) of a hypersphere

I am looking for a general formula for the Gaussian curvature of an $n$-sphere (the set of points in $R^{n+1}$ equidistant from the origin) of radius $r$. From what I have read, there would be $n$ ...
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1answer
202 views

“Completing” a vector field on a non-compact manifold $M$

Suppose I have a non-compact smooth manifold $M$ and an arbitrary nowhere-vanishing smooth vector field $X$ on $M$ which is not complete. Is there a way to create a smooth vector field $V$ that is ...
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1answer
245 views

Differential of smooth function on manifold

In the book I am using, the author defines differentials in the following way. Given smooth manifolds $M,N$ and a smooth mapping $\psi:M\to N$ define the differential $d\psi_m$ at a point $m\in M$ as ...
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2answers
572 views

What is the initial reason to define the evolute of a curve?

The evolute of a curve is defined as the envelope of the normals or as the locus of the center of the osculating circle. What is exactly "the envelope of the normals" ? What is the reason to ...
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1answer
252 views

Complete non-vanishing vector field

Let $M$ be a non-compact smooth manifold. Suppose we have a nowhere-vanishing smooth vector field X. Is this vector field complete? I know it is when $M$ is compact. However, I am unsure in the ...
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56 views

Cotangent bundle of a complex projective space

How does the cotangent bundle of a complex projective space looks like? Is that an Einstein manifold?
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252 views

Surfaces of Constant Gaussian Curvature

I'm preparing for an exam and I would like to know what are some examples of surfaces with constant Gaussian curvature such as surfaces with $k=0, \pm1$
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425 views

Implicit Function Theorem and Rank Theorem Misunderstandings.

Regular values are useful because of the generalization of the first part of the implicit function theorem: if $q$ is a regular value of $f:M \to N$ (with dimension $m$ and $n$ respectively), then $$A ...