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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sectional curvature of hyperbolic space

For a pair of $(X,Y)$ of linearly independent vectors in $T_pM$, $p\in M$, the sectional curvature is defined as $$K_p(X,Y)= - \frac{<R(X,Y)X,Y>}{|X|^2 |Y|^2 - <X,Y>^2}$$ The problem I'm ...
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Principal curvatures from parametrisation

Let $M^2$ be an immersed surface of the standard sphere $S^3$ with unit normal $\eta : M \to \mathbb{R}^4$ (tangent to $S^3$). Given a point $p \in M$ and a parametrisation $\varphi : U \subset \...
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Curvature of a hyperbolic plane

Consider a projective plane and a real quadric. According to the Klein-Beltrami-model the inside of the quadric is a hyperbolic plane. Klein proved that this plane has a constant negative curvature. ...
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Torsion and curvature of a linear connection

Could you help me to solve the following problem ? Let $M$ a parallelizable manifold of dimension $n$, {$E_1$,...,$E_n$} a global frame of $M$. Let $X$,$Y$ a vector fields on $M$ with $Y= \sum_{i=1}^...
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At what point does the curve y=(1/x) have max curvature? What happens to the curvature as x approaches infinity?

I have found kappa and kappa prime however have no clue where to go from here. Thanks Kprime = 6/-x^4(1+(1/x^4)^3/2-12/x^3(1+(1/x^4)^1/2) all over (1+(1/x^4))^3/2
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sectional curvature, ricci tensor and scalar curvature of the hyperbolic space [closed]

Who can help me to compute the sectional curvature, Ricci tensor and the scalar curvature of the hyperbolic space $H^3$ ? Thanks!!
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1answer
47 views

A Riemannian manifold with constant sectional curvature is Einstein. [closed]

A Riemannian manifold with constant sectional curvature is Einstein. Why? It's true the inverse?
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surface gradient of the unit normal vector

I am reading a book that defines a curvature tensor as $\boldsymbol{K} = -\nabla_\pi \boldsymbol{\hat{n}}$ where $\boldsymbol{\hat{n}}$ is the unit normal vector of a surface and \begin{align*} \...
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Raising to power as producing curvature

Is there some notion or theory that deals with the connection between the exponents of a polynomial and its curvature, i.e. how much it deviates from a straight line?
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Prove a curvature identity

Show that N = (dT/dt)/(abs (dT/dt)) using dT/ds = k N
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point of maximal curvature on $y=\ln x$

While attempting the problem below, I got the cross product of the function's derivative and second derivative to equal 0. If this happens, what does this mean for the curvature formula? My work: $$...
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Confusion on Gaussian curvature computation

Exercise I'm attempting to find the Gaussian curvature of the catenoid $M$ parametrized by $$ f(u,v)=(a\cosh v\cos u,a\cosh v\sin u,a v). $$ I've run through the typical computations of $E,F,G,e,f,g$ ...
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What is the difference between intrinsic and extrinsic curvature?

In general relativity, energy bends spacetime. However, this doesn't mean that a fifth dimension for spacetime to "bend into" exists." That is, spacetime isn't embedded in a higher dimensional space, ...
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Quick question: Curvature form of a connection on the trivial bundle

Let $L=\mathbb{R}^2\times U(1)$ be the trivial $U(1)$-bundle over $\mathbb{R}^2$. Define a connection $\nabla=d+A$ where $A=fdx+gdy$ is an $\mathbb{R}$ valued $1$-form on $L$. That is, $\nabla$ gives ...
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2answers
158 views

Reference of what metric can be placed on manifold?

I just read some conclusion that $T^2$ can't be placed metric with positive curvature at all points. I don't know why is so . And what book introduce about this ? I mean about what metric can be ...
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integration by parts on hypersurfaces

Usually the integration by part on the surface is trivial for planar domains. However, when it comes to hypersurfaces, some other terms like curvature show up. Can someone help with the understanding ...
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1answer
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Sectional curvature of 2-manifold

This is problem7, chapter 5 of Do Carmo's Riemannian geometry. Let $M$ be a Riemannian two manifold, $p\in M$ , $exp_p$ is a diffeomorphism on a neighbourhood of origin $V\in T_pM$. Let $S_r(0)\...
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How is the Euclidean mean curvature of a minimal submanifold of $ \mathbb{S^{n-1}} $ is equal to the metric Laplacian of the position vector?

I am reading about minimal cones from the book "A Course in Minimal Surfaces, T.H Colding, W.P Minicozzi II". It says that if $N^{k-1} \subset \mathbb{S^{n-1}}$ is $k-1$ dimensional minimal ...
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What will happen if evolve metric under Ricci flow on general manifold? [closed]

Because the scalar curvature under Ricci flow evolve by $$ \partial_t R=\Delta R+ 2|Ric|^2 $$ I treat it as heat equation with heat source . So, no matter the heat source is very hot or not , the ...
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Hypersurfaces of a hemisphere

Let $S_+^{n+1}$ be the open hemisphere of the standard euclidean sphere centered at the north pole and let $M^n$ be a compact, connected and oriented hypersurface of $S_+^{n+1}$. Is it true that if $M$...
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Sign disagreement in curvature form on $U(1)$-bundle

this is rather trivial question to masters, but as I'm totally disoriented by my computation, so deciding to ask. I'm trying to solve the following exercise, 1.1, showing the curvature 2-form $F(X,Y)=...
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1answer
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A Young's inequality used to bound curvature terms

I've been having a look at the Gage and Hamilton's The Heat Equation Shrinking Convex Plane Curves (here). In particular I've been working in the Lemma 4.4.2 and some further results where they find ...
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1answer
57 views

Fundamental group and curvature

Is there any paper about the $\pi_1$ group and curvature ? Because how close a curve depends on the curvature near the curve . I think there must have some condition which decide whether there is ...
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1answer
25 views

Curve with arc length have signed curvature k(s)>0?

Let $g:I \to \mathbb{R}^2$ be a curve such that for all $s \in I$, $\|g'(s)\|=1$ and $\kappa_g(s) \neq 0$, where $\kappa_g$ is the signed curvature of $g$. Is $\kappa_g(s) \gt 0$ for all $s \in I$? ...
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Find radius of curvature of a $3D$ surface at a point if the tangent vector at that point is given.

I have a $3D$ surface with equation $z=x^2 +y^2$ and need to find the radius of curvature at a point $P(x_1,y_1,z_1)$ using the tangent vector at that point. Since, there are many possible radii of ...
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Bianchi identity proof : why we can consider $[X,Y]=[Y,Z]=[X,Z]=0$?

I recall that the Riemann curvature tensor is defined by \begin{align*} R:\Gamma(M)\times \Gamma(M)\times \Gamma(M)&\longrightarrow \Gamma(M)\\ (X,Y,Z)&\longmapsto [\nabla _X,\nabla _Y]Z-\...
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curvature of planar hyperbola

If a planar hyperbola is parametrized as follows $$ x(t) = a\sec(t), \quad y(t) = b\tan(t), \quad a, b \text{ constants} $$ what is the curvature of the hyperbola curve?
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Computing Curvature of a Connection (Dirac Monopole)

I'm trying to verify some computations in a paper I'm reading and am feeling a little lost. In particular I haven't been able to properly compute curvature of a connection acting on a line bundle. ...
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How are the extended de Rham differential and the covariant derivative related by Cartan's 2nd structural equation?

I am reading Prof. N.Poncin's notes on fibre bundles and connections. You can find it here:https://orbilu.uni.lu/bitstream/10993/14274/1/MM4-9November2011.pdf In $\it{Section \ 4.5.2}$ the author ...
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Surface curvature for cylindrical jet for determining surface tension

enter image description here Hi, I have this problem of the stability of a cylindrical jet. The things I do not understand is the expression for the surface curvature for the cylindrical jet, which I ...
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How does the existence of a unique geodesic in non-positive curvature follow from Cartan-Hadamard?

Cartan-Hadamard states that a non-positively curved Riemannian manifold is covered by $\mathbb{R}^n$. How does it follow that in each homotopy class of paths from $x$ to $y$ there exists a unique ...
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Visual explanation of the indices of the Riemann curvature tensor

I'm trying to understand the meaning of the Riemann curvature tensor, but I don't seem to be ready yet to understand the detailed rigorous definition. Anyway, I managed to understand (this gif was ...
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Radius of $\mathbb{CP}^n$

The question I am asking is basically Is it possible/usual to define a "radius" for certain metrics on $\mathbb{CP}^n$ in analogy to the case for $S^n$? To provide some more context about what I ...
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Gauss curvature and bisectional curvature

Let $f:\mathbb CP^1\to X$ be an smooth embedding which its image is the curve $C$ where X is a Kahler manifold. Then is it correct that $$\sup_{\mathbb CP^1}K(f^*\omega)<\sup_C K(\omega)$$ where ...
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Frenet frame, curvature and torsion

Given a curve $\gamma:(-1,1)\to\mathbb R^3$ via $$\gamma(t):=(\frac{1}{3}(1+t)^\frac{3}{2},\frac{1}{3}(1-t)^\frac{3}{2},\frac{t}{\sqrt 2})$$ how can I find a) its Frenet frame, curvature and torsion? ...
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Constant curvature and torsion [duplicate]

How does one determine all space curves with constant curvature and torsion? And what happens if the torsion approaches to infinity while the curvature is a fixed constant? An idea would be much ...
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1answer
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Gaussian Curvature of a Pseudosphere.

I have been trying to find the gaussian curvature of a pseudo sphere. I assumed the parametrization: X(u,v) = (cos(u)*sech(v), sin(u)*sech(v), u - tanh(u)). I know that it's a surface of revolution ...
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when does a curve has a maximum point (it has to do with principle curvatures)

if K=1 and mean curvature both equal to 1, that means that i have an umbilical point right? then the maximum and minimum principle curvatures are equal, am i allowed to say that the curvature of a ...
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1answer
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Elastic curves - What is wrong about my solution?

Given a curve $\gamma:\mathbb R\to\mathbb R^2$ with $\Vert\gamma'\Vert=1$ and curvature $\kappa(s)=\frac{c}{\cosh s}$, $c\in\mathbb R$, how can I show that $\gamma$ is an elastic curve for some ...
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1answer
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Area of circle in terms of Gaussian curvature

I am asking about a formula in section 2 of these notes. Let $\rho|dz|$ be a conformal metric on $U\subset\mathbf C$. Then the Gaussian curvature of $\rho|dz|$ at $z\in U$ is defined as $$K_\rho(z)=-\...
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1answer
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Thoughts on Theorema Egregium

Due to this theory any mapping from the globe to a paper neccesairly have disortions. My question is there a theory which states the number of necessary map that gets this error down to a certain ...
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1answer
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Invariance of the rank of the trace of Riemannian curvature under a change of frame

Let $$ R=\begin{pmatrix} R_{11} & ... & R_{1n} \\ &...\\ R_{n1} &...& R_{nn} \end{pmatrix}, $$ where $R_{ikjl}$ is curvature tensor of a Riemannian manifold $(M,g)$ and $R_{ij}=...
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Proof of Euler's Theorem involving curvature.

Theorem: Let $φ$ be the angle, in the tangent plane, measured counterclockwise from the direction of minimum curvature $\kappa_1$ . Then the normal curvature $\kappa_n(φ)$ in direction $φ$ is given by ...
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Mean curvature flow - initial condition - mean-convex

The mean curvature flow of a surface given by a graph $X : B \subset \Bbb{R}^n \to [0,\infty)$ is given by $$ X_t (x,t) = H(x,t) \vec n(x,t) $$ where $H$ is the mean curvature and $\vec n$ is the ...
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A corollary of Li-Yau-Hamilton estimate

Picture below is from the Hamilton's The Harnack estimate for the Ricci flow .How to get the corollary 1.2 by Theorem 1.1 ? It seemly be not immediately and hard to compute. Maybe just because I am ...
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Find a differentiable curve on the paraboloid $\,z= 2x^2 + y^2$ with minimum curvature

Let $S$ be the graph of the function $\,f(x,y) = 2x^2+ y^2$ in $\,\mathbb{R}^3$ (a paraboloid with vertex at the origin). It is clear that $S$ is a regular surface which can be parametrized by the ...
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Software of symbolic computation

In Riemann geometry, there are many complex compute , for example in the picture below.If want to get 2.5.16 it needs about 3 page to compute. And it is easy to mistake because it is complex. But the ...
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1answer
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Relation between derivatives of chart, derivatives of unit normal and Gaussian Curvature

Don't know how the prove this apparently simple relation: if $x(u,v)$ is a chart of a surface $S$, with unit normal $N(u,v)$, then $N_u\times N_v=Kx_u\times x_v$, where $K$ is the Gaussian curvature. ...
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Given a first fundamental form, showing a particular second form cannot exist

If I have a first fundamental form $ \mathrm{d}u^2+\cos^2 u \mathrm{d}v^2$, I am trying to show that the second fundamental form cannot equal $f(u,v)\mathrm{d}v^2$ for a smooth function $f(u,v)$. I ...
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1answer
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curvature of an arc in S3, in stereographic projection

$r(t)$ is a unit 4-vector. The derivatives of $r$ are known and well-behaved. I'm interested in images of $r$ in stereographic projection – but (for purposes of this question) I don't yet know where ...