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 ...

learn more… | top users | synonyms (1)

1
vote
0answers
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 ...
0
votes
0answers
23 views

Riemann Tensor in Particular Frame

I'm trying to reproduce a calculation which requires computing the Riemann tensor in a particular frame specified by some vierbein $e_a$. I have a complete expression for the spacetime metric in some ...
3
votes
2answers
55 views

Does there exist higher degree graded derivations on $\Omega(M) $

Does there exist any other graded derivation on $\Omega(M)$ other than the one of degree one which is the exterior derivative (i.e. maps such as $d: \Omega^p(M) \rightarrow \Omega^{(p+r)}(M) $, where ...
0
votes
1answer
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 ...
0
votes
1answer
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 ...
2
votes
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 ...
3
votes
1answer
58 views

Star operator in the simplest form

Let $E$ together with $g$ be a inner product space(over field $\mathbb R$) , $\text{dim}E=n<\infty$ and $\{e_1,\cdots,e_n\}$ is orthonormal basis of $E$ that $\{e^1,\cdots,e^n\}$ is its dual ...
9
votes
1answer
124 views

gradient flow on $SU(n)$

Define the following cost functions $f_1, f_2 :SU(n) \rightarrow \mathbb{R}$ by $f_1(U) = Re \left( \text{Tr}\left(G^{\dagger} U \right) \right)$ and $f_2(U) = \left| \left( \text{Tr}\left(G^{\dagger} ...
0
votes
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 ...
1
vote
0answers
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 ...
0
votes
0answers
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 ...
2
votes
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 ...
2
votes
2answers
36 views

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 ...
7
votes
2answers
173 views

Mapping sphere surface to a vector space such that distances are preserved?

I have a unit radius sphere (say in 3D) centered on the origin. Thus the shortest distance between two points on the sphere is the geodesic. Is there a transformation (linear or non-linear) on the ...
1
vote
1answer
39 views

Two different definitions of a Liouville measure

Ok, I'm currently confused because of two different definitions for the Liouville measure associated to a smooth manifold $M$ of dimension $n$. These are: a) The measure $\mu$ on the cotangent bundle ...
0
votes
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 $0$ ...
0
votes
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) \, ...
2
votes
0answers
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: ...
1
vote
0answers
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). ...
2
votes
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?
3
votes
1answer
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 ...
3
votes
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 ...
1
vote
0answers
39 views

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 ...
1
vote
0answers
17 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 ...
2
votes
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 ...
2
votes
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 $I \subset R$ ...
6
votes
2answers
85 views

Master's Exploration in General Relativity

just throwing a query out to the Math community. I'm about to embark on a master's in Gravitation, Cosmology and General Relativity and was looking for possible subjects to start researching. My main ...
1
vote
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 ...
5
votes
2answers
90 views

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 ...
0
votes
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 ...
0
votes
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 ...
1
vote
0answers
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 ...
0
votes
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 ...
3
votes
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 ...
2
votes
0answers
20 views

Tubular neighborhood by restricting the Riemannian exponential map

Let $M$ be a Riemannian manifold (possibly non-compact, possibly non-complete) and $N\subseteq M$ a smooth submanifold (possibly non-compact). Does there exist a continuous $\mu\colon M\rightarrow ...
4
votes
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 ...
1
vote
0answers
28 views

$\alpha(t_0)$ orthogonal to $\alpha^\prime (t_0)$

Somebody asked this question a while back but only hints were posted--I want to have my entire proof examined. Let $\alpha(t)$ be a parametrized curve which does not pass through the origin. If ...
0
votes
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 ...
0
votes
1answer
18 views

Singular points of evolute and pedal

Let $\gamma$ be a unit speed plane curve $\gamma : I \to \mathbb R^2$. $T(s),N(s),k(s)$ are unit tangent vector, unit normal vector, curvature repectively. Evolute and pedal are defined as: ...
3
votes
0answers
59 views

Understanding tensors

Locally in a chart, a tensor field looks like $$T= T^{i_1,...i_n}_{j_1,...,j_m} dx^{j_1} \otimes...\otimes dx^{j_m} \otimes \partial_{i_1} \otimes ... \otimes \partial_{i_n},$$ where ...
0
votes
0answers
25 views

Is the dimension of a smooth manifold an invariant of the underlying set in it?

Let $M$ be a smooth manifold, $S$ a set and $f:M\to S$ a bijection (assuming of course, that such a function does exist). It's an easy exercise to show that $S$ can be given a differentiable ...
3
votes
1answer
34 views

Algebraic vs Geometric Picard groups

Given a smooth manifold $P$. I found two definition of a Picard group. The firs one is algebraic, $Pic(C^{\infty}(P)),$ defined as the set of self-equivalence bimodules (Morita equivalence) with ...
4
votes
0answers
111 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 ...
2
votes
1answer
42 views

How to reduce cubics in the plane to a canonical form?

I watched a video from Wildberger in the Differential Geometry series ( first, or third lecture, I don't remember ) where he says the following. The general format of a cubic curve is $$a x^3 + b ...
1
vote
1answer
59 views

Arc length function of a helix/spiral is convex?

Given the arc-length of a parametric curve, $\int_a^b\|\gamma'(t)\|$ if the parametric curve was non-convex, can the arc length be a convex function?If the parametric curve was convex, will the arc ...
1
vote
1answer
38 views

Closed form defines locally a function?

In my particular example, I have a volume form on a one-dimensional manifold, so i.e. this volume form $dx$ is closed. Now, I was wondering, does the integral function $f(y):=\int_{x_0}^{y} dx$ define ...
3
votes
1answer
45 views

Covariant and partial derivative commute?

I know that we have for a function $\Gamma: (-\varepsilon,\varepsilon)^2 \rightarrow M$ we have (at least I think I know that this is true) $$\nabla_{\frac{\partial \Gamma}{\partial s}} ...
9
votes
1answer
113 views

What is the Exterior Derivative Trying to Do?

$\newcommand{\R}{\mathbf R}$ Consider a smooth function $f:\R^n\to\R$ and let $Df:\R^n\to \R^{n*}$ be the map which takes a point $\mathbf a\in R^n$ to the linear map $Df_{\mathbf a}:\R^n\to \R$. ...
1
vote
0answers
40 views

Jacobi fields and variations

I want to show that every Jacobi field is a variation of geodesics, i.e. let $Y: I \rightarrow TM$ be a Jacobi field along a geodesic $\gamma$, then I want to show that $Y$ can be written as a ...