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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Induced metric via $\mathbb C P^n \cong SU(n + 1)/S(U(n) \times U(1))$

I was wondering if the homeomorphism above gives me the Fubini Study metric on $\mathbb C P^n$. More precisely: Consider $\mathbb C P^n$ equipped with the metric induced by the standard construction ...
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25 views

Curvature shortening flow for immersions

I was wondering if for an immersed curve in the plane, is it true that if the singularity points are evolved appropriately, then the curve becomes more embedded. And if so, would it eventually become ...
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21 views

Is the Zariski topology equipped with Eisenstein's metric an analytic submanifold?

Using $M=(C(\mathbb{R}),T_z)$ with the norm $(x,y) \to \log(\partial_x+\partial_y)$, we can easily define a derivative using distributions. I was wondering: Does this make $M$ an analytic manifold ...
3
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39 views

Geodesic vector field is well-defined

Let $(M,g)$ be a Riemannian manifold. I just learnt that for a curve $x:I\to M$ to be a geodesic, the geodesic equation $$\ddot{x}^k+\dot{x}^i\dot{x}^j\Gamma^k_{ij}=0$$ is equivalent to the condition ...
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15 views

Determining if an equation represents (?) a Riemann surface

This is my first exposure to Riemann surfaces. I have studied complex analysis in an introductory course, and spent the last few weeks learning a little bit of deeper theory with Conway's Functions of ...
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16 views

Show that if the Gaussian curvature and the mean curvature are identically zero all over $S$, then $S$ is a part of a plane in $\mathbb{R}^3$.

A parametrization $\phi(u,v)$ of a regular surface is called a 'conformal parametrization if $X_u . X_u$ = $X_v . X_v$ and $X_u . X_v=0$, i.e., $E=G$ and $F=0$. Let $S$ be a regular surface ...
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37 views

Whitney Embedding Theorem

This is very basic question, but from my previous question I learnt that "Whitney Embedding Theorems states that any smooth n dimensional manifold can be embedded in Euclidean space of dimension at ...
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1answer
7 views

Gradient in cylindrical coordinates

This is more of a maths question, but several sources point at different expressions for the gradient in cylindrical coordiantes. Sometimes I see the radial component for the gradient of a scalar ...
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21 views

Curvature shortening flow of embedded curves

QUESTION: I'm not sure how they proved part c in particular. Note that theorem 2.1 refers to Huiskan's distance comparison principle for evolving curves. I don't see why a separating boundary curve ...
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1answer
55 views

$SL(n)$ is a differentiable manifold

Prove that $SL(n)=\{A\in \Bbb{R}^{n\times n}:\det(A)=1\}$ is a differentiable submanifold. The determinant function is smooth since it's a polynomial, and we have $\det^{-1}(1)=SL(n)$. So it ...
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52 views

Flowout Theorem

I am reading Theorem 9.20 (Flowout Thoerem) from Lee's Introduction to Smooth Manifolds, Second edition. A part of the theorem states the following: Let $M$ be a smooth manifold and $S$ be a ...
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71 views

Is $M=\{(x,y)\in \Bbb{R}^2 : x^2 = y^5\}$ a differentiable submanifold?

Let $M=\{(x,y)\in \Bbb{R}^2 : x^2 = y^5\}$. Determine whether or not $M$ is a differentiable submanifold. I honestly couldn't get anything out of it. What is the standard approach to this ...
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112 views

Differences and similarities between Euclidean and Minkowski geometry

I am trying to get my head around the differences and similarities between Euclidean and Minkowski plane geometry. AS far as I understand it they are both affine geometries meaning the parallel ...
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1answer
32 views

Show that a nonconstant subharmonic function on a manifold cannot attain its supremum

PROBLEM: Suppose $f$ is a smooth non-constant function on a connected Riemann manifold $M$ of dimension 2 such that $f$ is bounded and $\Delta_M f \ge0$. Show $f$ cannot attain its supremum. I try ...
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1answer
26 views

Trigonometric parametrization of a genus g surface?

It is possible to find functions $\phi, \psi \in \mathbb{R}[sin(x), sin(y), cos(x), cos(y)]$, so that $S^2 = \phi( [0,1]^2)$ and $\psi( [0,1]^2)$ is a torus. Is it possible find, for any genus g, ...
3
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47 views

Hermitian metric on $L=\mathcal{O}_{\mathbb{P^n}}(1)$ over the projective space

Consider the line bundle $L=\mathcal{O}_{\mathbb{P^n}}(1)$ over the projective space. Locally is descriebd by $\{U_a,g_{ab}\}$ where $U_a=\{z_a\neq0\}$ is the standard covering of the projective ...
2
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1answer
29 views

Isothermic Surface

What is an Isothermic Surface intuitively? There are a couple of definitions, but I really don't understand what it means if a surface is isothermic. What are ist properties, what is it used for?
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2answers
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Differential equation for finding closest point on surface.

Inspired by this question I got to think about a more general case. Say I have any discretized surface and want to find closest point from each point outside of surface to the surface. Say that I can ...
2
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1answer
42 views

About the definition of regular surface in do Carmo

According to do Carmo, the definition of regular surface requires us to check $X^{-1}$ to be continuous (where $X$ is a local parametrization). But doesn't it infer from other conditions (as shown in ...
3
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1answer
232 views

Writing a parametrization of the cissoid by using $\theta$

The cissoid of Diocles is the curve whose equation in terms of polar coordinates $(r,\theta)$ is $$r = \sin\theta \tan\theta, −\frac{\pi}{2}<\theta <\frac{\pi}{2}$$ Write down a parametrization ...
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54 views

Difference between exponential maps composed with parallel transport along two different geodesics?

Let $(M,g)$ be a Riemannian manifold, and let $\gamma_{p,v}, \gamma_{p,w}$ be two geodesics starting from $p$ with directions/initial vectors $v,w$ respectively. Consider the two operations (to be ...
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21 views

Symplectic form on a Hilbert Space is Closed

Let $\mathcal{H}$ be a Hilbert space over $\mathbb{C}$. Define a new vector space, $V$, over $\mathbb{R}$, which has, on the level of sets, $V = \mathcal{H}$ and for scalar multiplication (only with ...
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1answer
54 views

Show that $f^*\omega = \det(df) \, dx_1\wedge\cdots\wedge dx_n$

Let $f:\Bbb{R}^n\to \Bbb{R}^n$ be a differentiable map given my $f(x_1,\ldots,x_n)=(y_1,\ldots,y_n)$, and let $$\omega=dy_1\wedge\cdots\wedge dy_n.$$ Show that $$f^*\omega = \det(df) \, ...
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1answer
32 views

Are $n$-dimension cubes $C^k$ manifolds with boundaries?

I had a look at this very helpful post: What does it mean to say a boundary is $C^k$? From what I can understand higher dimension cubes do not fall in this category, because there is no way we can ...
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1answer
35 views

Heat kernel formula on hyperbolic plane well defined

Consider the heat kernel for the hyperbolic plane $\mathbb{H}^2$ and the corresponding heat kernel: $$k(x,y,t)=\frac{C}{t^{\frac{3}{2}}}\cdot ...
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1answer
24 views

Prove that $s: B \rightarrow E$ is a section ( vector bundles)

I'm very very unfamiliar with vector bundles, so maybe this question is quite trivial. Let $\pi : E \rightarrow B$ be a vector bundle and $s: B \rightarrow E$ a map sending each $p \in B$ to the $0$ ...
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18 views

Criteria for boundary convexity of hypersurfaces in Euclidean space

I have a question about the relationship between two different formulations of the notion of boundary convexity, in the sense of Riemannian geomety. Let $M$ be an $n$-dimensional manifold with ...
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2answers
451 views

Some questions on hyperelliptic compact Riemann surfaces

For genus > 1 hyperelliptic Riemann surface the definition guarantees that there is a degree 2 map from that to $\mathbb{P}^1$. Under this map the inverse image of the "point at infinity" has to be ...
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2answers
119 views

convex curve equivalent definition

I have seen two different version of definition of a closed convex curve in a plane: For a curve $r(t)=(x(t),y(t))$ 1.The whole curve lies on only one side of any tangent of the curve 2.Any ...
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Why does it suffice to check the geodesic equation to leading order?

I am reading Taubes's book on differential geometry and am wondering about a proof. My apologies if this is simple, as I'm still grappling with the material. My question concerns material in chapter ...
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26 views

The heat equation shrinking convex plane curves

In the article "The heat equation shrinking convex plane curves" by M. Gage and R. S. Hamilton, I didn't understand the following theorem: ...
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37 views

The curvature blows up

A curve evolves according to the evolution equation $\displaystyle\frac{\partial X}{\partial t}= k \times N$ where $k$ is the curvature of the curve and $N$ is the inward unit normal vector. Then ...
3
votes
2answers
82 views

Curvature flow for convex planes curves

Tentative translation of the original question. I've read several articles on the curvature flow for convex plane curves (the curve remains convex during evolution, and eventually shrinks to a point). ...
3
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1answer
104 views

Is it generally difficult to memorize 'multivariable calculus' theorems?

There are many weak forms of "Fubini's theorem" with strong hypotheses in elementary calculus texts. However, these strong hypotheses are very unnatural and are thus hard to memorize. Compared to ...
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Given an embedded $C^k$ submanifold of $R^n$, is the distance function $C^k$?

I am asking because I am wondering if studying smooth manifolds is the same as studying zero loci of smooth functions with non-vanishing gradient. Let me be a little formal for clarity: Let $M$ be a ...
3
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1answer
20 views

What is meant by a “curvature-line parametrization” of a surface?

Could anyone explain to me what it means if a surface is curvature-line parametrized? What does it mean intuitively and how exactly is it different from any other parametrization? I've been looking ...
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1answer
285 views

Review on Riemannian Geometry

I'm currently reading through Griffiths and Harris Principles of Algebraic Geometry, and the only subject in the foundational material section that I am not completely comfortable with is riemannian ...
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42 views

Expansion of path-ordered integral and curvature

Suppose $(P,\pi, M)$ is a principal bundle with structure group $G$ and suppose $\omega \in \Omega^1(P,\mathfrak{g})$ is a connection on $P$ with curvature $\Omega = D\omega$. If $\sigma : U\subset M ...
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1answer
36 views

$\alpha(t)\cdot v=0$ for all $t$

Let $\alpha:I\rightarrow R^3$ be a parametrized curve and let $v\in R^3$ be a fixed vector. Assume that $\alpha^\prime (t)$ is orthogonal to $v$ for all $t\in I$ and that $\alpha(0)$ is also ...
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38 views

Connection giving an identification of the double-tangent bundle

Let $M$ be a manifold, and let $\nabla$ be a (possibly torsion-free) connection on $M$. Then I am pretty sure that $\nabla$ induces an isomorphism between the double-tangent bundle and the Whitney sum ...
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1answer
50 views

How does the steepness of lines through a hyperboloid change the further away they are from the apex?

I'm a geoscientist and am trying to figure out how the slopes of the flanks of a hyperboloid change for straight lines that cross them. The line of reference is through the apex (red). Now take any ...
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2answers
31 views

Maximal geodesics on compact manifolds

I have two questions about the following passage in Taubes's book on differential geometry. I also quote the proposition it references. 9.1 The maximal extension of a geodesic Let $I \subset R$ ...
3
votes
1answer
171 views

What is the geodesic equation on $\mathbb{S}^{n}$?

Suppose $\gamma: \mathbb{R}\rightarrow \mathbb{S}^{n}$ is a smooth curve. Let $\gamma(t)=(x^{1}(t)...x^{n+1}(t))$. Let $\mathbb{D}^{n}$ be embedded into $\mathbb{R}^{n+1}$ by viewing ...
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1answer
34 views

Showing a vector field is smooth.

Let $(M,g)$ be a Riemannian manifold, $N$ a smooth manifold and $$\pi:M\to N$$ a surjective smooth submersion. Then, each level set $M_q=\pi^{-1}(q)$ is a properly embedded submanifold of $M$ so we ...
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1answer
39 views

Notation: determinant of Jacobian matrix

Given a function $f:\Delta^n \to Y$ from a simplex into a riemmannian manifold. Furthermore given a point $x \in \Delta^n$ we can send an orthonormal basis at $x$ using $D_x f$ to a set of vectors ...
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2answers
31 views

Tangent vectors as derivations and dot product

Take $\mathbb{R}^n$ with the Euclidean metric to be the manifold of interest for this question. Suppose we have two vectors $v_p = v^i (\partial/\partial x^i)_p$ and $w_p = w^j (\partial/\partial ...
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2answers
650 views

Equation of a straight line in spherical coordinates

I'm trying to prove the angle sum formula for a triangle on the surface of a sphere. In order to do this I wanted to create a general triangle on the sphere, with one vertex at $\theta = 0$ and one ...
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28 views

Scaling of minimal surfaces

After scaling and suitable Euclidean motions every rigid minimal patch can be placed on a unit catenoid of revolution $ x^2 + y^2 = c^2 \cosh^2 (z/c), c=1.$ with full area contact. Is the statement ...
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81 views

Vector valued 2-forms which satisfy Jacobi Identity

Motivated by this MO question we ask the following two questions: 1)What is an example of a compact manifold $M$ which does not admit any smooth (1,2) tensor $\omega$ which restriction to each ...
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1answer
14 views

Smooth Conjugate Net vs. Curvature-Line Parametrization

so I was wondering what a smooth conjugate net exactly is, intuitively? Also, what exactly is a curvature-line parametrization? What would it mean that a smooth conjugate net is orthogonal? Why is it ...