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)

learn more… | top users | synonyms

1
vote
0answers
13 views

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? ...
1
vote
0answers
10 views

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 ...
0
votes
1answer
28 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 ...
0
votes
0answers
8 views

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 ...
0
votes
1answer
41 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 ...
0
votes
1answer
39 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 ...
2
votes
1answer
56 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 ...
0
votes
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 ...
0
votes
1answer
39 views

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 ...
3
votes
0answers
18 views

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 ...
0
votes
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 ...
3
votes
0answers
22 views

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 ...
0
votes
0answers
27 views

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 ...
1
vote
1answer
11 views

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. ...
3
votes
0answers
25 views

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 ...
2
votes
1answer
15 views

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 ...
0
votes
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} \\ ...
0
votes
1answer
21 views

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 ...
0
votes
1answer
16 views

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. ...
1
vote
0answers
36 views

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 ...
1
vote
1answer
25 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 ...
1
vote
0answers
23 views

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 ?
-1
votes
1answer
27 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 ...
0
votes
0answers
13 views

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 $$ ...
0
votes
0answers
26 views

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 ...
1
vote
1answer
34 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 ...
0
votes
0answers
11 views

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 ?
2
votes
2answers
35 views

Why image of curvature is a Lie subalgebra?

In the red line of picture below, why it is Lie algebra ? $M_{\alpha\beta}$ is the Lie bracket ? But $M_{\alpha\beta}$ is symmetric . Picture below is from the 216 page of this paper. ...
0
votes
0answers
25 views

Proof of Hamilton's strong maximum principle.

As picture below, Why $\forall v\in \text{null}(M_{\alpha\beta}), \nabla_iv\in\text{null}(M_{\alpha\beta})\Rightarrow \text{null}(M_{\alpha\beta}) \text{ is invariant under parallel translation}$ ? ...
0
votes
0answers
29 views

Rank of curvature operator under Ricci flow.

I think under Ricci flow ,the rank of curvature operator does not change by +1 or -1, it will directly change to full rank or zero rank . I want to write it as term paper, but I don't know whether ...
0
votes
1answer
30 views

The value of the integral of the curvature of a complex line bundle

I am trying to show that if $L \to M$ is a complex line bundle endowed with a connection $\nabla$, $F$ is the curvature form, and $S \in M$ a closed surface, then $\int \limits _S F \in 2 \pi \textrm ...
0
votes
1answer
33 views

Geometric quantization: not understanding the curvature form and Weil's theorem

I am reading a bit about geometric quantization. The texts that I am following are N.M.J. Woodhouse's "Geometric Quantization" and an article from arXiv. I am having troubles understanding the ...
1
vote
0answers
20 views

Well-determined tangent in the complex plane

I am trying to understand a theorem that involves curvature in the complex plane. The author of the theorem makes the statement that a smooth function $K : T \rightarrow R$ is the curvature of a ...
3
votes
1answer
53 views

No conjugate points on $S^1\times \Bbb R$

Lee claims in his book that $S^1\times \Bbb R$ (considered as a submanifold of $\Bbb R^3$) admits no conjugate points along any geodesic. I am struggling to make that rigorous. Being conjugate along ...
1
vote
1answer
53 views

The radius of curvature of a surface of revolution

The radii of curvature $R_1$ and $R_2$ at point $A$ are $R_1=AM$ ($M$ is the center of curvature of the meridian curve in the plane of the figure) and $R_2=AN$ (in the perpendicular plane). $\vec{n}$ ...
2
votes
1answer
49 views

Eigenvalues of shape operator and of curvature on second exterior power

Terminology note In the following, a scalar product will be a symmetric bilinear form, and a euclidean scalar product will be a positive definite scalar product. This is the terminology used by my ...
3
votes
0answers
35 views

Example of a complex manifold with certain curvature properties?

Are there (nice) examples of a complex manifold such that the sectional curvatures through all the complex planes are non-positive but the sectional curvatures through the real planes are mixed?
1
vote
1answer
29 views

If $M_{\alpha\beta}\ge0$ , how to show $M^\#\ge0$?

We say $M_{\alpha\beta}\ge0$ if $M_{\alpha\beta}v^\alpha v^\beta\ge0$ (sum over $\alpha,\beta$) for all vectors $v=\{v^\alpha\}$. If $M_{\alpha\beta}\ge0$ ,how to show $$ M^\#\ge0 ~? $$ Relative ...
2
votes
0answers
33 views

Total Gaussian curvature

For a compact surface, $S$, in $\mathbb{R}^3$, how would I go about showing that the total Gaussian curvature $\int_S K da \leq 4 \pi$? I feel like Hopf's Umlaufsatz and the Gauss-Bonnet Theorem are ...
2
votes
1answer
36 views

Computing Sectional Curvature on Hyperbolic Plane

For a pair of $(X,Y)$ of linearly independent vectors in $T_pM$, $p\in M$, the sectional curvature is defined as $$K_p(X,Y)=\frac{<R(X,Y)Y,X>}{|X|^2 |Y|^2 - <X,Y>^2}$$ The problem I'm ...
3
votes
1answer
40 views

Laplacian of a distance function on a Riemann manifold

For some reasons I need to show the following fact. Let $(M, g)$ be a Riemannian manifold. Let $U \subset M$ be an open set and $r: M \to \mathbb{R}$ a smooth distance function. Let us assume ...
0
votes
1answer
19 views

Existence and uniqueness of a point with horizontal tangent in a convex curve

I had a look on the proof by E. Schmidt of the Schur's Theorem about arcs of convex curves. It states the following: Let $C$ and $C'$ be two arcs of the same length with the endpoints $a,b,a',b'$ ...
0
votes
1answer
32 views

Can a function have curvature if it has no inflection point

The function $$f(x) = \dfrac{1}{x^2-1}$$ has no inflection point, but can it have curvature (concave/convex) ?
0
votes
2answers
22 views

Determine the condition makes the curve helix

(Twisted cubic) Show that $\alpha(t)=(at,bt^2,ct^3)$ is a helix $\iff$ $4b^2 = 9a^2c^2$. I think $\frac{\tau}{\kappa}$ must be constant. I've solved for them. However, I got really mixed equations ...
1
vote
0answers
36 views

Gaussian curvature of $S^3$

It is easy to see that Gauss curvature of $S^2$ is $1/R^2$. How can we find the Gaussian curvature of $S^3$? What about $S^n$ in general?
1
vote
1answer
42 views

Gaussian curvature versus sectional curvature

I was studying https://en.wikipedia.org/wiki/Gaussian_curvature (exact version https://en.wikipedia.org/w/index.php?title=Gaussian_curvature&oldid=709607678 ) and there it says: (bold added) ...
4
votes
1answer
45 views

Derive $\kappa(t)=\frac{\lvert \boldsymbol{a}(t) \times \boldsymbol{v}(t) \rvert}{v^3(t)}$ directly instead of proving it.

Let $\boldsymbol{r}(t)$ be a parametrised curve. Then, $$\boldsymbol{\hat{T}}=\frac{\boldsymbol{r'}(t)}{\lvert \boldsymbol{r'}(t) \rvert }$$ ...
1
vote
1answer
32 views

Counting independent components of Riemann curvature tensor

I'm having some trouble understanding the counting procedure for the number of independent components of Riemann curvature tensor $R_{iklm}$ in 4D spacetime. (The answer is supposed to be 20, but I'm ...
1
vote
1answer
32 views

Finding the curvature and torsion of a curve

For this question: I'm not able to find what the torsion is for the curve gamma. In my notes I'm given that first derivative of b, (b dot) =-tau multiplied by n. so in the solution I don't ...
0
votes
0answers
18 views

find the point which principle curvature has max value.

$z(x,y)=\sum\limits_{m=1}^{\infty}\sum\limits_{n=1}^{\infty}\dfrac{16q_0}{(2m-1)(2n-1)\pi^6D}\bigg[\dfrac{(2m-1)^2}{a^2}+\dfrac{(2n-1)}{b^2}\bigg]^{-2}\times\sin{\dfrac{(2m-1)\pi ...