# Tagged Questions

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### Monotonic curvature and self intersections.

I'm trying to prove that if $\alpha: I \to \Bbb R^2$ is a differentiable curve, where $I$ is an interval, has strictly monotonic curvature, then $\alpha$ has no self-intersections. My attempt: We ...
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### The curvature of a Cycloid at its cusps.

My lecturer proposed a question to particular result regarding the curvature of a Cycloid (generated by circle of radius 1) at its cusps. Having left it as an open problem, I thought it'd be ...
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### Curve and Constant Curvature

I have initial position vector $p_0$, given curve-linear length $1$. It can be parameterized by $s\in[0,1]$. Assume we have the equation to generate the curve from given starting point and constant ...
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### Constant curvature geodesic circles on a surface with constant Gauss curvature

Referring to: Curvature of geodesic circles on surface with constant curvature, Is it possible to combine further the last two of the three equations in the link given above into a single ODE / PDE ...
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I'm trying to prove the formula to calculate the curvature of a plane curve. But I end up with the wrong sign and can't figure out why: I want to proof $\kappa(t) = \frac{\dot c(t) \cdot \ddot ... 2answers 60 views ### Ricci Tensor, Curvature and Scalar Curvature computation from definition I am studying a little of Riemannian geometry and I am having some problem in making the connection between two expressions of Ricci tensor, curvature and scalar curvature. Well, in the book that I am ... 1answer 32 views ### Easy solution to Yamabe problem for surfaces The Yamabe problem asks if, given a Riemannian manifold$(M,g_0)$, it is possible to find a conformal metric$g$on$M$with constant scalar curvature. I would like to know if there is some "easy" ... 0answers 44 views ### Shape operator and orthogonality of eigenvectors When studying differential geometry (at a hobby level) I always run into problems when it comes to the varying notations and statements about the shape operator ... 0answers 34 views ### Finding curve that minimizes an integral due to constraints In the euclidean plane I want a smooth curve$\gamma (t)$which satisfy:$$\gamma (0)=(0,0)\quad\quad \gamma '(0)=(1,0)\quad\quad n\in \mathbb{N}\land n\neq 1\Rightarrow \gamma ^{(n)}(0)=(0,0)\\\gamma ... 2answers 71 views ### Global and local coordinates on a manifold, and their relations to curvature I would be pleased to have some information about coordinates in differential geometry. A) First I would like to check whether or not the definitions I use are correct. (Mainly for the sake of ... 2answers 58 views ### Angle between two vectors on manifold I'm parallel transporting a vector along a curve and trying to calculate how much this vector rotates relative to the curve's tangent vector. So if the path is a geodesic then I will get an answer of ... 1answer 42 views ### Proving that a surface is isometric to the plane A surface S has first fundamental form du^2 + G(u,v)dv^2 and curvature 0. Also the curve u=0 is a geodesic when parametrized by arclength. Prove that G(u,v) = 1 i.e. that S is isometric ... 1answer 46 views ### Calculate the curvature of a space curve at the point M(-1,5,-4) In order to calculate the curvature for this space curve, do I use this formula? And where does the point M come into this? P.S. This is probably a silly question, but I'm new to differential ... 1answer 71 views ### Radius of curvature for the plane curve x^3 + y^3 = 12xy. Could someone help me with this problem? : Determine the radius of curvature for the plane curve x^3 + y^3 = 12xy at the point (0, 0). 1answer 91 views ### Prove the Gaussian curvature K=0 If two families of a geodesics on a surface intersect at a constant angle \theta, prove that the Gaussian curvature of the surface is zero, i.e. K=0. Please explain how to show the question. ... 1answer 60 views ### Which of principal curvature, Gaussian curvature and mean curvature is intrinsic ？ Why? Which of principal curvature, Gaussian curvature and mean curvature is intrinsic and why? Which book will have this explanation? I read a book (Remannian Geometry, Do Carmo, p129). It has a saying: ... 2answers 106 views ### Where does this expression of Gaussian curvature come from? In my Differential Geometry course, we have seen a way to calculate the Gaussian curvature K given a metric expressed as the sum of two Pfaff forms Q = ω_1^2 + ω_2^2: we find another Pfaff form ... 2answers 63 views ### Is a tangent to a curve in a hyperbolic plane straight? Consider a projective plane with an absolute quadric, so that it is a hyperbolic plane. Given a curve I wonder how the tangent to a curve is defined in a plane with constant positive curvature. I ... 1answer 44 views ### Gaussian curvature of a parallel surface Question 11 section 3.5 in Do carmo part c.Let a surfae x have constant mean curvature equal to c does not equal 0 and consider the parallel surface to x at a distance 1/2c. Prove that this parallel ... 2answers 65 views ### Computing the Gaussian curvature of this surface z=e^{(-1/2)(x^2+y^2)}. Compute the Gaussian curvature of z=e^{(-1/2)(x^2+y^2)}. Sketch this surface and show where K=0 , K>0, and K<0. So would the easiest way to do this question be to construct a ... 1answer 20 views ### Mean curvature an asymptotic directions I'm a little confused by this question I've come across in do carmo while studying for my final. (Sect 3.2 #7) Show that if the mean curvature is zero at a nonplanar point, then this point has two ... 2answers 61 views ### Gauss Curvature of a Surface Find the Gauss curvature of a surface with the first quadratic form:$$\mathrm{d}s^2 = \mathrm{d}u^2 + 2\cos a(u,v)\mathrm{d}u\,\mathrm{d}v + \mathrm{d}v^2.$$I have found E, F, and G. E = ... 0answers 42 views ### Curvatures of 4-Dimensional Parallel Curve Let \vec{\alpha}: I \rightarrow \mathbb{R}^4 be an arc-length parametrized curve in \mathbb{R}^4 with curvatures k_1, k_2, k_3. The principal normal unit vector is \vec{n} = ... 1answer 53 views ### Flat Surfaces in \mathbb{R}^3 Can Be Bent Only Along Straight Lines This is a problem out of Elementary Differential Geometry by Barrett O'Neill (Chapter 6 Section 3 Number 2). Let M be a flat surface in \mathbb{R}^3 with principal curvatures k_1 and k_2, ... 0answers 27 views ### Scalar Curvature of a metric on the hemisphere, from a paper on the Min-Oo Conjecture I'm reading a paper on the Min-Oo Conjecture (http://arxiv.org/abs/1004.3088), and I'm stuck on the following step in a proposition: Given a metric g_0(t) on the upper hemisphere \mathbb{S}^n_+, ... 0answers 40 views ### Frenet formulas for curves in arbitrary Riemann manifold As far as I understand, to have Frenet formulas one would need a curve, embedded in \mathbb{R^n} and, desirably, naturally parametized. But there are homonomic notions of curvature and torsion of ... 1answer 37 views ### Why use Gauss and mean curvature to characterize a surface's deviation from being “flat” at one point? We know for a 2-dimensional surface there are two orthogonal principal directions at every point, where the principal curvatures \kappa_1 and \kappa_2 are the two ends of the curvature spectrum ... 3answers 93 views ### Finding the Total Curvature of Plane Curves I'm trying to find the total curvature (or equivalently, rotation index, winding number etc.) of a plane curve (closed plane curves) given by$$\gamma(t)=(\cos(t),\sin(nt)), 0\leq t\leq 2\pi$$for each ... 0answers 52 views ### Proof check for critical point definition with mean curvature I'm currently trying to prove: "Definition: We say that a surface S \in R^3 is minimal if it is a critical point for the area functional" Starting with this: "If we consider a family of smooth ... 1answer 55 views ### Cartan formalism calculation Just to test out the Cartan formalism, I decided to apply it to the sphere. So, it admits a metric,$$\mathrm{d}s^2 = \mathrm{d}r^2 + r^2 \sin^2 \phi \mathrm{d}\theta^2 + r^2 \mathrm{d}\phi^2$$from ... 2answers 87 views ### Total/Gaussian curvature is intrinsic, yet mean curvature is extrinsic, why? What characteristics define the total/mean curvature to be intrinsic/extrinsic accordingly? What is different geometrically about these curvatures that cause them to be defined as this? 1answer 54 views ### Two curvature formulas when equal arc-length all. So with a parametric curve \vec{r}=\langle x(t),y(t)\rangle, curvature is given by$$\kappa=\frac{|x'y''-x''y'|}{(x'^2+y'^2)^{3/2}}.$$When we have constant arc-length, an alternate ... 1answer 55 views ### Curvature of saddle by definition I'm trying to compute the principle curvatures of the saddle M defined by z= y^2 -x^2 at the point p = (0,0,0), but I know my computations are wrong. Maybe you can help to see where I went ... 0answers 51 views ### Curvature tensor on the sphere While reading R. Hamilton paper "Three-manifolds with positive Ricci curvature" I came across the sentence: For the sphere we have$$R(u,v,u,v)=R_{ijkl}u^{i}v^{j}u^{k}v^{l}>0,$$which is the ... 1answer 120 views ### Relations between curvature and area of simple closed plane curves. Let \gamma be a simple closed plane curve. We know that a curve with constant curvature \kappa will trace a circle in the plane. The radius of this circle is the inverse of its curvature. Now, ... 0answers 22 views ### How can I retreive Cartesian positions from curvature and torsion? I have a set of 3D points in the Cartesian space. What I want is to extract the TNB (frenet frames) of these points together with the curvature and torsion. Which seems to be easy. Then I want to do ... 1answer 309 views ### Riemann tensor in terms of the metric tensor The Riemann tensor is a function of the metric tensor when the Levi-Civita connection is used. Most tensor notation based texts give the Riemann tensor in terms of the Christoffel symbols, which are ... 0answers 87 views ### Question about spherical curvature ( binormal, tangent vectors) Let a:I\mapsto R^3 be a unit speed curve. if \rho^2+(\rho'\sigma)^2 is constant and equal to r^2, show that a curve is on a sphere with r radius. answer is: define \gamma:I\mapsto R^3 ... 1answer 97 views ### Trying to understand Gauss-Bonnet theorem Update: I was able to figure this out on my own. The problem was that I assumed the \phi coordinate runs from 0 to 2\pi, but the metric so defined has a conical singularity. To eliminate the ... 0answers 134 views ### The curvature and torsion of the tangent indicatrix Let \alpha be a unit speed curve. Its tangent indicatrix \sigma is defined by \sigma(t)=T(t). Find torsion and curvature of \sigma with respect to the torsion and curvature of \alpha. ... 2answers 130 views ### when can you estimate curvature from finite information about two geodesics? Let c_v, c_w be two geodesics starting at a point p\in M, where M is a nonpositively curved, complete, smooth Riemannian manifold. Say c_v(\varepsilon) = \exp_p(\varepsilon v) and ... 1answer 44 views ### gaussian curvature in isotherm parameterization Consider an isothermic parameterization with metric$$(g_{ij}) = \begin{bmatrix} \lambda^2 & 0 \\ 0 & \lambda^2\end{bmatrix}$$with \lambda = \lambda(u^1,u^2)>0. Then the gaussian ... 2answers 289 views ### Ricci curvature: step in proof of a paper by Hamilton In Hamilton's paper "The Ricci Curvature Equation" (in Seminar on Nonlinear Partial Differential Equations, here), I can do all of Lemma 4.2 except for the following relation:$$ ... 1answer 33 views ### If$dX_1 = dX_2$then curvatures of$\nabla^{X_1}$and$\nabla^{X_2}$agree Let$E \simeq M \times \mathbb C$be a trivial smooth complex line bundle over the Riemann surface$M$and let$S \colon M \to E$be its smooth nowhere vanishing section. Let$\nabla^1$and$\nabla^2$... 1answer 36 views ### Let$c:I\rightarrow\mathbb R^3$a regularly parametrized curve with curvature$\kappa=0$where$I\subset\mathbb R$is an interval. Show that the image of$c$is contained in a straight line. we defined the curvature with;$\kappa_c(t)= ...
I am not terribly well-versed in differential geometry, so please keep that in mind when answering the question. We are given a surface in ${R}^3$ defined parametrically by $\vec{r}(u,v)$ where ...