In differential geometry, the term curvature tensor may refer to the Riemann curvature tensor of a Riemannian manifold, the curvature of an affine connection or covariant derivative (on tensors), or the curvature form of an Ehresmann connection. (Def: http://en.m.wikipedia.org/wiki/Curvature_tensor)

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Math Rap fact check

I'm a high school math teacher and I write math raps every year for my students. I'm working on my lyrics and I need help making sure something is mathematically accurate. I'd like to make reference ...
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Mean and Gaussian curvature - normalization to interval $[0, 1]$ [on hold]

I can compute curvature of the $2.5D$ surface. Problem is, I need the results scaled in interval $[0,1]$ (or $[-1,1]$). Is it possible to compute this directly or I need to compute all curvatures of ...
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Schur's theorem in DoCarmo's “Riemannian Geometry”

The exercise 8 of chapter 4 of Do Carmo's "Riemannian Geometry" ask to prove the Schur's Theorem. I don't understand a step in the hint (the "hint" is essentially the proof of the theorem). Schur's ...
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15 views

Possibility of regular surface with specific first and second fundamental form matrices

I have met this in diff. geometry class which states: We are to determine if there exists a regular surface in $ R^3 $, $ S = f(u,v) $ with fundamental forms as follows: $ I = \begin{bmatrix} ...
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Curvature Relationship with Norm of the Curve at the Point of Maximum Norm

Continuing on my series of questions, for those following, is the following question: Let $\alpha: I \to \mathbb{R}^3$ be a regular curve. Suppose that for some $t \in I$ the distance from the ...
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16 views

Conformal curvature line parametrization

While reading a paper I found a definition which is confusing me. Def: A conformal curvature line parametrization $(x,y) \to F(x,y)$ is called isothermic. I know what a conformal ...
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13 views

Help with understanding a proof of compact surface having an elliptic point

In my studies of differential geometry from do Carmo's book, I have come across a very nice claim which states that a regular compact surface has an elliptic point that is a point with positive ...
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29 views

How to measure curvature of a curve using slope?

I am analyzing a financial instrument that has seven prices. The seven prices each have an expiration of 1 month, 2 month, 3 month etc. I would like to measure the curvature of the curve. I know how ...
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An alternative derivation of radius of curvature (2D functions). How valid is it?

I was wondering how radius of curvature was derived, and this is what I came up with. It turned out to be longer than expected. Then I looked at how it compares with other (presumably more ...
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Geodesics on surfaces of revolution about z axis with negative curvature

This is a question in differential geometry of surfaces that I could not do We are given S a surface of revolution about the z axis with everywhere negative Gaussian curvature. We are to show that the ...
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Find the versors of the Frenet Triad

I have the following curve: $C(t)=(\frac{t^5}{5}, \frac{t^3}{3}, t ^ 2).$ The problem is to find $T$, $N$ and $B$ of the Frenet Frame. I know the fact that $\vec{B} = \vec{T} \times \vec{N}$. I've ...
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Geodesics on surface of revolution of regular curve

I was recently presented with this in differential geometry stating the following: Let us define the regular curve on the XZ plane as: $ \gamma (t) = (sin(t)+2,0,t) $ on XZ plane for $ t \in R $, ...
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How to parameterize these pretty hyperbolic surfaces?

I've seen the attached images describing surfaces of negative curvature. I was wondering if there exist such surfaces with constant Gaussian negative curvature. To this end, I attempted to model the ...
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45 views

Angle between position and velocity vectors is constant?

Is there a name for such a curve or can this even happen? I know when the velocity vector, $\mathbf{x'}$, and position vector, $\mathbf{x}$ are always orthogonal $\mathbf{x}(t)$ parametrizes a circle ...
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29 views

Differential Geometry Proof Regarding Arclength, Tangents, Curvature, and Parameters

Consider a regular curve q(t) with arclength parameter s. Show that if $T(t_{n}) \neq T(t_{0})$ and $t_{n} \rightarrow t_{0}$, then $$1 = lim_{t_{n} \rightarrow t_{0}} \frac{|\theta(t_{n}) - ...
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Connection with zero torsion and curvature

I remember encountering the following theorem many, many time ago: if a connection $\nabla$ has $R = 0$ and $T = 0$ then... The problem is that I cannot remember the conclusion. Was it that the ...
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1answer
30 views

How to understand it will sweep out a 2-dim manifold?

As red line in picture below, I really can't imagine why it will sweep out a 2-dim submanifold. Below picture is from the 9th page of John M. Lee's 'Riemannian Manifolds: An Introduction to ...
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Formula for the curvature $2$-form.

I'm currently reading a textbook to do with curvature and $k$-forms. It says that the curvature $2$-form given connection $1$-form, $A$, is $$F =d^A A = dA+A \wedge A$$ It then goes on to say that ...
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41 views

The parameter curves are asymptotic curves

I am looking at the following exercise: Let $p$ be a hyperbolic point of a surface $S$. Show that there is a patch of $S$ containing $p$ whose parameter curves are asymptotic curves. Show that ...
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parallel postulate of Euclidean geometry and curvature

In elementary geometry, we have two standard examples which violate the (strong) parallel postulate of Euclidean geometry: in hyperbolic geometry, we have more than one parallel through a point which ...
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1answer
47 views

An “application” of Gaussian curvature

I'm trying to help a collegue in his differential geometry exam. His exam assignment is to discuss gaussian curvature and some of its properties and applications. I know that the gaussian curvature ...
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Sectional curvature in a paraboloid is always positive.

I'm working on Lee's book ''Riemaniann Manifolds an Introduction to Curvature''. One exercise (11.1) is about to see that the paraboloid given by the equation $y=x_1^2+...+x_n^2$ has positive ...
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A calculate of curvature connecting intuitional images

When I read this Wiki (picture below is from it), it's so great that explain what the Riemann curvature tensor is.But I get stuck in the calculate in the red box. How to calculate it ? Because ...
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62 views

How do we get the terms $E, \ F, \ G, \ L, \ M, \ N$?

I am looking at the following exercise: Show that a curve $\gamma (t) = \sigma (u(t), v(t))$ on a surface patch $\sigma$ is a line of curvature if and only if $$(EM − FL) \dot u^2 + (EN − GL) \dot u ...
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How to show $\nabla s(x)$ and $\nabla^2 s(x)$ are bounded? [closed]

Let $(M,g)$ is smooth complete Riemann manifold with bounded curvature. $exp$ is exponent map.For some given point $x_0\in M$ ,let $$ s(x)=d(x_0,exp_x(v)) $$ $d$ is distance function,$v\in T_xM$, and ...
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A specific example about computing of curvature.

After learned some Riemann Geometry, I want to compute the curvature of $S^2$ by the way of Riemann curvature. So, I assume the $S^2$ is $$ (\theta_1,\theta_2)\rightarrow ...
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36 views

Hessian comparison

I get stuck in the red line in picture below. First, although $\varphi$ is smooth, but seemly the derivertive about $x$ can't be moved to $\varphi$.So, I don't know why it's smooth. Second,I want to ...
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110 views

Geodesic curvature of sphere parallels

I want to compute the geodesic curvature of any circle on a sphere (not necessarily a great circle). $$$$ The geodesic curvature is given by the formula $$\kappa_g=\gamma'' \cdot (\textbf{N}\times ...
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34 views

Compute the curvature of $g = ds^2 + \frac1M \sinh^2(\sqrt M s) d\Omega^2$?

About 1.5 month ago, I ask this question . Today I think I can resolve it.Because I choice to think the easy case $n=3$.And in this case, I know what is the $d\Omega^2$, I'm confident. So , I do it . ...
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Curvature identity on Nearly Kähler manifolds

Can somemone help me prove the following identity? $$ \| \nabla_X(J)(Y) \|^2 = \langle R_{XY}X,Y\rangle - \langle R_{XY}JX,JY\rangle$$ where $J$ is the almost complex structure, and $R$ the curvature ...
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21 views

Centres of Curvature

Suppose we have a space curve $r(t)=\left(x(t), y(t), z(t)\right)$ and we consider the locus of all centres of curvature of $r$. First, would this be the correct formula? $$r(t) + {1 \over \kappa(t)} ...
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Path on Unit Sphere for which Binormal is Parallel to Position

I am wondering if it is possible to produce a path on the unit sphere for which the Binormal vector ${B}$ is parallel to the position vector at some instant (or all instants!). I tend to think of the ...
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1answer
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Geometric Interpretation for Torsion

This is actually two questions about the Wolfram MathWorld article on torsion here. 1: What is a geometric interpretation of torsion? For curvature I understand it as the reciprocal of the radius of ...
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Curvature tensor for a particular Hilbert manifold

My question involves an infinite dimensional Hilbert manifold with a Riemannian metric. My question is: What is the form of the curvature tensor for a infinite dimensional Hilbert manifold with ...
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2answers
27 views

Intution: second derivative corresponds to curvature?

What's the intuition for understanding that the second derivative does correspond to the curvature of the function? For first derivatives one can think of the tangent lines, but what to think for ...
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Question about the normal bundle of a totally geodesic submanifold of a Cartan-Hadamard manifold

I'm reading the exposition in Lang's Fundamentals of Differential Geometry of the following generalization of the Cartan-Hadamard theorem: Suppose $X$ is a Cartan-Hadamard manifold (i.e. a complete, ...
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Do torsion and curvature have higher order analogues?

Consider the usual formulas for the Torsion and Curvature of an affine connection: $$T(X,Y)=\nabla_X Y -\nabla_Y X-[X,Y]$$ $$R(X,Y)=\nabla_X\nabla_Y-\nabla_Y\nabla_X-\nabla_{[X,Y]} $$ These formulas ...
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Ricci curvature with positive lower bound…

There is a theorem of Reilly concerning the first Dirichlet eigenvalue $\lambda$ of the Laplacian on a n-dimensional compact Riemannian Manifold $(M,g)$ whose Ricci curvature $Ric$ has a positive ...
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Curvature measure for polygones on a 2D space

I would like to implement a curvature measure for polygones on a 2D space. My goal is to compute shape parameters to know if the polygon is close to a circle or has a sinuous shape or an elongated ...
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What is the mean of $R(v_{ij})$

As picture below ,I can compute $\delta R=-v_{jl}R_{jl}+\nabla_i\nabla_lv_{il}-\Delta v$.But in the last line (1.5.2), what is the mean of $R(v_{ij})$? I think $R$ is a function on manifold, why the ...
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Can the scalar curvature remain locally constant under the action of the Ricci flow?

Disclaimer: I know close to nothing about differential geometry, only basics due to my early interest in general relativity when I was in high school almost 20 years ago. Therefore the present ...
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2answers
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Is a Riemannian manifold with isometric coordinate charts flat?

Suppose $M$ is a Riemannian Manifold such that each point has a coordinate chart into $\mathbb{R}^n$ that is an isometry, in the sense that the inner products are preserved. Does this imply that $M$ ...
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Relation between curvature and sectional curvature

Let $(M,g)$ be a Riemannian manifold and $ h = c.g$ for some $c > 0$ . Then the Levi-Civita connections of $g$ and $h$ are same. From the above deduce the relation between corresponding curvature ...
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How could we calculate the signed curvature?

We have that $$\epsilon (s)=\gamma (s)+\frac{1}{\kappa_s(s)}\cdot n_s(s)$$ We regard $\epsilon$ as the parametrization of a new curve, called the evolute of $\gamma$ (if $\gamma$ is any regular ...
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Compact Einstein manifolds with $\operatorname{Ric}(g)=\lambda g$ with $\lambda<0$ and sectional curvatures $\geq0$

Does there exist a compact Einstein manifold $(M,g)$ with $\operatorname{Ric}(g)=\lambda g$ and $\lambda<0$ and nonnegative sectional curvatures?
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Curvature on product Riemannian manifolds

I am working on the following problem from Lee's Riemannian Manifolds: Suppose $g = g_1 \oplus g_2$ is a product metric on $M_1 \times M_2$ (i.e. $$g(X_1+X_2,Y_1+Y_2) = g_1(X_1,Y_1)+g_2(X_2,Y_2),$$ ...
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Show that a developable surface has zero Gaussian curvature.

A developable surface is a ruled surface $x(s,t)=\alpha(s)+t\beta(s)$, where $\alpha(s)$ is a unit-speed curve and $|\beta(s)|=1$, such that the tangent planes along each ruling are parallel. Show ...
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1answer
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Finding dual forms of a frame field on a sphere

I'm attempting to calculate the Gaussian curvature of the sphere of radius $r$, but I'm not sure how to find the dual forms of the frame field. I start with the parametrization $X(\phi, \theta) = (r ...
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Counterexample to the form of Gromov compactness theorem without a Ricci curvature bound

Gromov compactness theorem states that in a class of Riemannian manifolds that have a uniformly bounded diameter and uniformly bounded below Ricci curvature every sequence of manifolds has a ...
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A curve of constant curvature and zero torsion must be a circle

From Elementary Differential Geometry by Pressley I don't understand the last paragraph. Why does it show $\gamma$ lies on the sphere $\mathcal S$ with center $\mathbf a$ and radius $1/\kappa$?