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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Embed an $n\times n$ matrix into $R^{n^2}$

How to embed a matrix, for example, a $3x3$ singular matrix in to $R^9$? How to compute the induced metric? Is it just the Frobenius norm of the matrix? Many Thanks. sam
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298 views

Help with definition of n-dimensional smooth manifold

Again, I am reading this. I am finding it a bit difficult to understand the definition of n-dimensional smooth manifold. Now, $\{U_a; x^1_a, x^2_a, ..., x^n_a\}$ ...
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1k views

How do I compute Gaussian curvature in cylindrical coordinates?

I just asked this question on ask.metafilter, and it was suggested that I ask here. Though I'm talking about coding something up, this question is about the math behind it, not the implementation. ...
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Why is the Möbius strip not orientable?

I am trying to understand the notion of an orientable manifold. Let M be a smooth n-manifold. We say that M is orientable if and only if there exists an atlas $A = \{(U_{\alpha}, \phi_{\alpha})\}$ ...
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531 views

Quick question on Riemannian geometry

I got a quick question on riemannian geometry. I'm not quite sure whether this is the right place to ask this question, since it might be a rather elementary one from a research point of view. ...
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What are some examples of $\text{Isom}(M)$ and $\text{Conf}(M)$?

Edit: Since I did not quite get the responses I would have liked when I asked this question four months ago, let me reformulate it slightly: What are some examples of $\text{Isom}(M)$ and ...
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260 views

Approximating the volume of the Jacobian of a hyperelliptic curve

For an abelian variety $A_{/\mathbb{Q}}$, its volume $vol(A(\mathbb{R}))$ appears in the conjectured Birch Swinnerton-Dyer formula for the L-series at 1. I am having trouble in understanding the size ...
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famous space curves in geometry history?

For an university assignment I have to visualize some curves in 3 dimensional space. Until now I've implemented Bézier, helix and conical spiral. Could you give me some advice about some famous ...
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735 views

Integrating a $k$-dimensional (multivariate) Gaussian over a convex $k$-polytope

What is the integral of a $k$-dimensional (multivariate) Gaussian over a convex $k$-polytope? Here I am specifically interested in $k\in\{2,3\}$, but insight on the general problem would also be ...
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1answer
889 views

Cross product of operators

How to show that: $ (-i\nabla-eA)\times(-i\nabla-eA) = (ie\nabla \times A) $ i and e are constants A is a vector field $\nabla$ = vector differential operator
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432 views

If $g^{-1} \circ f \circ g$ is $C^\infty$ whenever $f$ is $C^\infty$, must $g$ be $C^\infty$?

Suppose that $g$ is a bijection on the real line, and $g^{-1} \circ f \circ g$ is a $C^\infty$ function whenever $f$ is $C^\infty$. It seems howlingly obvious that this can only happen if $g$ is ...
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Adding handles to a sphere

I am trying to work through Boothby's An Introduction to Differentiable Manifolds on my own and, embarassingly, have got stuck at the very first chapter. At the end of section 4, chapter 1 (called: ...
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1answer
181 views

Some questions about geometry on pseudo-Riemannian manifolds

I will be using relativistic terminology for pseudo-Riemannian manifolds as in the book by Barret O' Neil. If one can show that the chronal future and the chronal past of some set are disjoint then ...
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1answer
524 views

Locus of osculation of concentric ellipses (elliptic pond ripples)

If you dropped two rocks in a pond, the concentric circles emanating from the two spots would osculate $\infty$ times. The locus of osculating points would form a line. Now imagine that instead of ...
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Does a Fourier transformation on a (pseudo-)Riemannian manifold make sense?

the Fourier transformation of a scalar function with respect to one variable might be defined as $\mathcal{F}\left[w\right](\omega )\equiv ...
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2answers
362 views

Connections and differential equations

I was trying to understand the notion of a connection. I have heard in seminars that a connection is more or less a differential equation. I read the definition of Kozsul connection and I am trying to ...
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486 views

Intuition about geodesic incompleteness

To state the context, I am familiar with the Hopf-Rinow theorem. My request is three fold, I would like to know of general classes of geodesically incomplete spaces. I basically want to see lots ...
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157 views

What is the Gauss part of Gauss-Manin connection?

The definition of Gauss-Manin connection involves de Rham cohomology. Surely, Gauss didn't work with de Rham cohomology as we know it. So, what was the context in which Gauss came up with this idea?
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Teaching myself differential topology and differential geometry

I have a hazy notion of some stuff in differential geometry and a better, but still not quite rigorous understanding of basics of differential topology. I have decided to fix this lacuna once for ...
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189 views

Parametric Equations for a Hypercone

The n-dimensional cone, with vertex at the origin, central angle, $\alpha$ and central axis in the direction of the unit vector $\xi$ is defined to be all those points, $x\in {R^n}$ whose dot product ...
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3answers
519 views

How to prove a manifold is simply connected?… using geometry

I was Looking at another questions title, and given the tag of DG, I thought it would read a little more like this one. Or at least that answers to this question would be answers to that question. ...
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2answers
571 views

Why study “curves” instead of 1-manifolds?

In most undergraduate differential geometry courses -- I am thinking of do Carmo's "Differential Geometry of Curves and Surfaces" -- the topic of study is curves and surfaces in $\mathbb{R}^3$. ...
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Tensors as matrices vs. Tensors as multi-linear maps

So I read the answers in this question, and don't feel that much closer to an answer about how tensors as multi-linear maps and tensors as "multi-dimensional" matrices are truly related. For ...
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1answer
271 views

Prove using an admissible unit speed curve and a Frenet frame

Assume $f:(a,b) \to \mathbb R^3$ is an admissible unit speed curve (hence $f^{\prime} \times f^{\prime\prime}$ is never zero) If $f$ lies on the sphere with center $a$ and radius $r$ prove that $f ...
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1answer
207 views

Admissible curve contained in a plane

(Assume a unit speed parametrization) Prove that an admissible curve $c:(a,b) \to \mathbb R^3$ (hence $c^{\prime} \times c^{\prime\prime}$ is never zero) with a zero torsion is contained in a plane. ...
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173 views

Unit Normal Field on a 2D manifold embedded in R^3

Let us assume that we are given a closed, orientable 2D manifold embedded in $R^3$, and lets call it $M$. I think it is clear that in a coordinate neighborhood $(U, \phi)$ it is possible to at each ...
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2k views

Is there an easy way to show which spheres can be Lie groups?

I heard that using some relatively basic differential geometry, you can show that the only spheres which are Lie groups are $S^0$, $S^1$, and $S^3$. My friend who told me this thought that it ...
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3answers
514 views

Supremum length of space curves contained in the open unit ball having always less than unity curvature

I am in the process of proving that if a space curve (in R^3) has infinite length and the curvature tends towards 0 as the natural parameter s tends to infinity, the curve must be unbounded - i.e. not ...
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1answer
306 views

What is the name of the matrix used to weight an inner product?

In Linear Algebra, when computing an inner product $<x,y> = y^*Wx$, what is the name of the matrix W? If it doesn't have a name, where can I find a practical explanation of how to construct ...
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1answer
179 views

What Re(f(z))=c is if f is a holomorphic function?

Suppose that $f:U\subset\mathbb{C}\to\mathbb{C}$, where $U$ is a region in the complex plane, is a holomorphic function. If $c\in\mathbb{R}$ is a regular value for $\text{Re}(f(z))$ then it follows ...
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2answers
1k views

Differential and Riemannian structure on the cone

I think the cone (or what is also called the "half cone") is a differential manifold but not a smooth manifold. Can anyone help me understand this the nuts and bolts way? How explicitly can I ...
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755 views

Computing the Chern-Simons invariant of SO(3)

I am an undergraduate learning about gauge theory and I have been tasked with working through the two examples given on pages 65 and 66 of "Characteristic forms and geometric invariants" by Chern and ...
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270 views

Geodesic on half-plane determined by tangent vector

The upper-half plane $\mathbb H$ carries a hyperbolic metric and the geodesics are semicircles with base on the real line. We consider oriented geodesics. Let $x \in \mathbb H$ and let $v$ be a unit ...
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Special types of Sasaki manifolds

i have a question to special cases of Sasaki-manifolds. Let $(M, g, \xi, \eta, \Phi)$ a Sasaki-manifold. In special case maybe $M=S^{2m+1} \cong \mathbb{C}^{m+1}$. This is a Sasaki manifold. But what ...
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4answers
6k views

Why is a circle in a plane surrounded by 6 other circles

When you draw a circle in a plane you can perfectly surround it with 6 other circles of the same radius. This works for any radius. What's the significance of 6? Why not some other number? I'm ...
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3answers
689 views

How to prove a manifold is simply connected

A manifold $M$ is simply connected if for every pair of 1-cubes $c_1,c_2: [0,1]\rightarrow M$ with $c_1(0) = c_1(1) = c_2(0) = c_2(1) = t$ there is a 2-cube $b$ such that 1) $b(1,0) = c_1$ and $b(1,1) ...
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183 views

How do you show that the Laplacian is the square of the (Euclidean) Dirac operator?

If I understand correctly, the Euclidean Dirac operator is given by $$D=\sum_{i=1}^n e_i \frac{\partial}{\partial x_i},$$ where $e_i$ are bases for $Cl_{0,n}(\mathbb{R})$, i.e., the $n$-dimensional ...
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3answers
587 views

Where do we need the axiom of choice in Riemannian geometry?

A friend of mine is a differential geometer, and keeps insisting that he doesn't need the axiom of choice for the things he does. I'm fairly certain that's not true, though I haven't dug into the ...
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722 views

Volume of a geodesic ball

This may be embarassingly simple, but I can't see it. Let $M$ be a Riemannian manifold of dimension $n$; fix $x \in M$, and let $B(x,r)$ denote the geodesic ball in $M$ of radius $r$ centered at $x$. ...
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253 views

Geodesics on Compact Manifolds

Let $M$ be a compact, connected smooth manifold. If $p, q$ are points in $M$, is there always a geodesic that goes from $p$ to $q$? I know that this is certainly not true if $M$ is not compact, but I ...
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1answer
349 views

Orientation on a Manifold

Let M be an (n-1)-manifold in R^n . Let M(e) be the set of end-points of normal vectors (in both directions) of length e and suppose e is small enough so that M(e) is also an (n-1)-manifold. Show ...
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2answers
950 views

Expression for Levi-Civita Connection

I'm having trouble with the following exercise in do Carmo's Riemannian geometry. Let $X$ and $Y$ be differentiable vector fields on a Riemannian manifold $M$. Let $p \in M$ and let $c: I \to M$ be ...
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1answer
519 views

Question on Partitions of Unity

I was reading John Lee's Introduction to Smooth manifolds, and I came across this question: Let $M$ be a smooth manifold, and let $\delta : M \rightarrow \mathbb{R}$ be a positive continuous ...
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287 views

To what extent do the stories on manifolds carry over to schemes?

This is a follow-up (refinement?) of this question. In learning some algebraic topology, I've learned to think of an affine scheme as spec $R$. (I've been told that this is a legitimate use of ...
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744 views

Manifold interpretation of Navier-Stokes equations

I am wondering about particle trajectories for solutions of the Navier-Stokes equation. Is it possible that there is a Manifold $M$ for which fluid particles move along geodesic's or "straight lines ...
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5answers
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Why the interest in locally Euclidean spaces?

A lot of mathematics as far as I know is interested in the study of Euclidean and locally Euclidean spaces (manifolds). What is the special feature of Euclidean spaces that makes them interesting? ...
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2answers
880 views

PDEs on Manifolds

I am wondering if there is a general coordinate-independent way to define a Partial Differential Equation on a Smooth manifold. It is definitely true that in each coordinate neighborhood you could ...
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0answers
859 views

The set of symmetric matrices as a manifold

How would I start off proving that $S= $(the set of symmetric $n\times n$ matrices) is a manifold. I tried using the definition directly by saying $M_n =$ the space of all $n\times n$ matrices For ...
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2answers
175 views

Harmonic functions on $S^2$

Consider the sphere $S^2 = \lbrace (x,y,z) :\ x^2 + y^2 + z^2 = 1 \rbrace$. This is a smooth manifold in $\mathbb{R}^3$, and for a given point $s \in S^2$, one can consider its coordinate ...
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3answers
1k views

The function that draws a figure eight

I'm trying to describe a counterexample for a theorem which includes the figure eight or "infinity" symbol, but I'm having trouble finding a good piecewise function to draw it. I need it to be the ...