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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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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Reference about the surgery of Ricci flow

I roughly read the Topping's LECTURES ON THE RICCI FLOW. Seemly, there is not introduction about surgery. Seemly,it is enough to deal singularity by blow up. Then, for knowing surgery I read the ...
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2answers
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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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33 views

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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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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42 views

What will happen if evolve metric under Ricci flow on general manifold?

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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19 views

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
38 views

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
54 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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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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44 views

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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35 views

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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69 views

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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Extrinsic curvature for cylinder

Suppose we have the following metric describing a cylinder: $$ds^2=ud\rho^2+\rho^2d\phi^2$$ where $u$ is a function of $\rho$. We know the definition of the extrinsic curvature that is, $$K_{ij}=-\...
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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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11 views

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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45 views

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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42 views

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
34 views

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
44 views

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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42 views

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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59 views

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
19 views

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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1answer
34 views

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 ...
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1answer
50 views

Whether there is easy way to compute $R_{ij}=\frac{1}{2}Rg_{ij}$ in 2-dimension

In 2-dimensional Riemann manifold ,Ricci curvature is given by $$ R_{ij}=\frac{1}{2}Rg_{ij} $$ My PDE teachers teach me to compute it by the way. $$ R_{11}=g^{ij}R_{1i1j}=g^{22}R_{1212} \\ R_{12}=g^...
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Why the well-defined of Gauss map depends on surface is orientable?

Let $S$ is a surface. Define a mapping $g:S\rightarrow S^2\subset R^3$ of $S$ into the unit sphere $S^2$ , associating to every $p\in S$ a unit vector $N(p)\in S^2$ normal to $T_pS$. Why the well-...
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Calculate curvature of wave

I am looking for a way to calculate curvature of this wave (pic attached) in matlab. Sinc Wave I have generated this wave form in Matlab. ...
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How to show $\nabla_a\nabla_c R_{ad}-\nabla_c \nabla _a R_{ad}=-R_{cade} R_{ae}-R_{ce} R _{de}$?

$\nabla$ is Riemann connection and $R_{ij}=g^{kl}R_{ikjl}$. How to show $\nabla_a\nabla_c R_{ad}-\nabla_c \nabla _a R_{ad}=-R_{cade} R_{ae}-R_{ce} R _{de}$ ? Or generate commutator of generate ...
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1answer
26 views

finding curvature radius

given a projectory equation of the form $ y=y(x) $find the curvature radius as a function of $x.$ a projectory equation , hence $ x=x(t)$, input that in y and we get $y=y(x(t))$, which is what one ...
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Metric evolving under Ricci flow with nonnegative scalar curvature is shrinking?

Let $g_{ij}(x,t)$ be a complete solution of the Ricci flow on a surface with bounded and nonnegative scalar curvature ,why the metric is shrinking ?
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29 views

Are curvature forms in complex line bundles symplectic

I know that the curvature form $F_\nabla$ of a connection $\nabla$ in a complex line bundle $L \to B$ is presymplectic (i.e. antisymmetric and closed). Does it also have to be non-degenerate, i.e ...
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How to Compute Discrete Intrinsic Curvature

Given a function $f$, a region $S$ where this function is twice differentiable, and the property that all of its discrete difference series centered in $S$ converge within $S$ We have that $$ f''(x)...
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Computing of proof of Li-Yau estimate

I try to compute the red line in picture below: \begin{align} \Delta(\partial_tL) +R\Delta L +\partial_t R &=\Delta(Q+|\nabla L|^2)+R(\frac{\Delta R}{R}-\frac{|\nabla R|^2}{R^2}) + \Delta R +R^2 ...
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1answer
38 views

Parallel surface

For a regular surface $\mathbf{x} = \mathbf{x}(u,v)$ Define $\mathbf{y}(u,v) = \mathbf{x}(u,v) + t \mathbf{N} (u,v)$ where $\mathbf{N}$ is the unit normal of $\mathbf{x}$ How could I show the ...
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Invariant of support function and support point under parallel translation

Picture below is from the 222 and 220 page of this paper,why the support function and support point is invariant under parallel ?