# Curvature Definition and 42 Discussions

In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.
For curves, the canonical example is that of a circle, which has a curvature equal to the reciprocal of its radius. Smaller circles bend more sharply, and hence have higher curvature. The curvature at a point of a differentiable curve is the curvature of its osculating circle, that is the circle that best approximates the curve near this point. The curvature of a straight line is zero. In contrast to the tangent, which is a vector quantity, the curvature at a point is typically a scalar quantity, that is, it is expressed by a single real number.
For surfaces (and, more generally for higher-dimensional manifolds), that are embedded in a Euclidean space, the concept of curvature is more complex, as it depends on the choice of a direction on the surface or manifold. This leads to the concepts of maximal curvature, minimal curvature, and mean curvature.
For Riemannian manifolds (of dimension at least two) that are not necessarily embedded in a Euclidean space, one can define the curvature intrinsically, that is without referring to an external space. See Curvature of Riemannian manifolds for the definition, which is done in terms of lengths of curves traced on the manifold, and expressed, using linear algebra, by the Riemann curvature tensor.

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1. ### I Is there a program that can solve for Gaussian curvature of a Metric?

I've been trying to find a way to calculate Gaussian curvature from a 4D metric tensor. I found a program that does this in Mathematica using the Brioschi formula. However, this only seems to work for a 2D metric or formula (I would need to use something with more dimensions). I've found...
2. ### I I have a question about the mathematical model of a reflective curve

Hello : Have a question regarding the mathematical model of reflective curve where could i find information on it ? (pdf , webpages , ebooks ,...etc ) Other than Wikipedia Best Regards HB
3. ### A Section 5.1 of Wald's *General Relativity*

From the section[5.1] of 'Homogeneity and Isotropy' from General Relativity by Robert M. Wald (pages 91-92, edition 1984) whatever I have understood is that - ##\Sigma_t## is a spacelike hypersurface for some fixed time ##t##. The hypersurface is homogeneous. The metric of whole space is ##g##...
4. ### I Global coordinate chart on a 2-sphere

Hi, I know there is actually no way to set up a global coordinate chart on a 2-sphere (i.e. we cannot find a family of 2-parameter curves on a 2-sphere such that two nearby points on it have nearby coordinate values on ##\mathbb R^2## and the mapping is one-to-one). So, from a formal...
5. ### I Visualization of Spatial Curvature - Schwarzchild Solutions - Interior & Exterior

I'd love have a little discussion about the Interior Schwarzschild Solution. Here's a diagram I slapped together to illustrate the key points. (I assume everyone reading this familiar with embedding diagrams, and using an axis to 'project' a value, in this case the spatial z-axis is replaced by...
6. ### Proving ##C## is constant in 4-dim ##R_{\mu\nu}=Cg_{\mu\nu}##

This question wasn't particularly hard, so I assume metric compatibility and input Ricci tensor to the left side of Einstein's equation. $$R_{\mu\nu}-\frac{1}{2} Rg_{\mu\nu}=Cg_{\mu\nu}-\frac{1}{2} (4C)g_{\mu\nu}=-Cg_{\mu\nu}$$ Then apply covariant derivative on both side...
7. ### Exercise 16, chapter 3 (Tetrad) in Carroll

My attempt at solution: in tetrad formalism: $$ds^2=e^1e^1+e^2e^2+e^3e^3≡e^ae^a$$ so we can read vielbeins as following: \begin{align} e^1 &=d \psi;\\ e^2 &= \sin \psi \, d\theta;\\ e^3 &= \sin⁡ \psi \,\sin⁡ \theta \, d\phi \end{align} componets of spin connection could be written by using...
8. ### A Parallel transport of a 1-form aound a closed loop

Good day all. Since the gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve. Then If we form the Gradient vector field...
9. ### A Different FRW Cosmological Models

ρο/a4 ρο/a3 k Λ Βig Bang Big Crunch Polynomial Expansion Exponential Expansion x 0 x 0 >0 x +1 x +1 x 0 <0 x -1 x -1 >0...
10. ### Riemann Curvature Tensor in 2D

Since in 2D the riemman curvature tensor has only one independent component, ## R = R_{ab} g^{ab} ## can be reversed to get the riemmann curvature tensor. Write ## R_{ab} = R g_{ab} ## Now ## R g_{ab} = R_{acbd} g^{cd}## Rewrite this as ## R_{acbd} = Rg_{ab} g_{cd} ## My issue is I'm not...
11. ### I Fourier transform on manifolds

Is there a generalized form of the Fourier transform applicable to all manifolds, such that the Fourier transform in Euclidean space is a special case?
12. ### A Fundamental definition of extrinsic curvature

My question is quite simple: what is the fundamental definition of extrinsic curvature of an hypersurface? Let me explain why I have not just copied one definition from the abundant literature. The specific structure on the Lorentzian manifold that I'm considering does not imply that an...
13. ### I Stress-energy tensor contribution to curvature

Hi everyone. Could you help me to find the way to prove some things? 1)Changing of body velocity or reference frame don't contribute to spacetime curvature 2)On the contrary the change of body mass causes the change of curvature in local spacetime I use the assumption that if we have the same...
14. ### I Curved Spacetime for E&M

So General Relativity explains the force of gravity as mass/energy induced curvature of spacetime. This correctly predicts gravitational time distortion, nonlinear geodesics and gravitational lensing, the anomalous precession of planetary orbits, the schwarzchild metric, and so on. Could the...
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### I What properties allow me to see past Earth's curvature?

I recently was able to view a 193 foot building from 24 miles away. The base of the building is approximately 15 feet above sea level and my eye level was approximately 9 feet above sea level. I was viewing the building across a Lake. I could see a substantial amount of the building, which...
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### B How to do the calculations showing the Universe is flat?

I've been trying to understand how we know that the observable universe is flat, and I'm having difficulty finding any sources that explain exactly how the calculations were done. On this WMAP website (https://map.gsfc.nasa.gov/mission/sgoals_parameters_geom.html), it says: "A central feature of...
17. ### B Mechanism of Curvature of Space

Hi all, How does matter curve space (what's the mechanism)? Does this Curvature happen instantaneously or does it happen at light speed? Thanks in advance.
18. ### I Non-diagonal metric for testing elliptic PDE solver

Hello, I am working with numerical relativity and spectral methods. Recently I finished a general elliptic PDE solver using spectral methods, so now I want to do Physics with it. I am interested in solving the lapse equation, which fits into this category of PDEs  \nabla^2 \alpha = \alpha...
19. ### I Understanding Parallel Transport

I'm currently in a GR class and have come across the notion of parallel transport, and I've searched and searched the last few days to try and understand it but I just can't seem to wrap my head around it, so I'm hoping someone here can clarify for me. The way I picture parallel transport is...
20. ### I Mean and Gaussian curvature for a Gaussian 'hill' seem wrong

I'm hoping someone can help check whether my final contour plots look plausible based on the surface. I haven't done too much differential geometry but I've needed to work with Gaussian/Mean curvature for a simple 3D gaussian surface. Here's an example: (A = 7, a=b=1/(3.5)^2) It's...
21. ### Show curvature of circle converges to curvature of curve @ 0

Homework Statement Let γ : I → ℝ2 be a smooth regular planar curve and assume 0 ∈ I. Take t ≠ 0 in I such that also −t ∈ I and consider the unique circle C(t) (which could also be a line) containing the 3 points γ(0), γ(−t), γ(t). Show that the curvature of C(t) converges to the curvature κ(0)...
22. ### Show reparameterized curvature equals curvature up to a sign

Homework Statement Let γ: I → ℝ2 be a smooth regular curve and let λ = γ ο φ with φ: Iλ → I be a reparameterisation of γ. Show, by using the general formula for curvature of a regular curve that κλ = ±κ ο Φ where the ± depends on whether φ is orientation preserving (+) or reversing (-)...
23. ### A Interpretation of the derivative of the metric = 0 ?

I am trying to learn GR, primarily from Wald. I understand that, given a metric, a unique covariant derivative is picked out which preserves inner products of vectors which are parallel transported. What I don't understand is the interpretation of the fact that, using this definition of the...
24. ### Osculating Circle

Homework Statement I know I would use the curvature equation |f''| / [1-(f')^2]^3/2 and then take the limit of that to -1. I just don't understand why I have to take the limit of the curvature and when I take the limit of the curvature I get |-1| / (13)^3/2 when the answer should be 2.
25. ### B Observational limit from the super-horizon mode spectrum

In making cosmological measurements, we are limited to the region within the particle horizon, the 'observable universe'. However, it is reasonable to assume that even if the universe is finite, it is much larger than that volume. If, for example, we measure the curvature ##\Omega_K##, the value...
26. ### Finding the curvature of a space curve

Homework Statement Find the curvature of the car's path, K(t) Car's Path: r(t) = \Big< 40cos( \frac {2 \pi}{16}t ) , 40sin( \frac {2 \pi}{16}t ), \frac{20}{16}t \Big> Homework Equations K(t) = \frac { |r'(t)\:X \:r''(t)|}{|r'(t)|^3 } The Attempt at a Solution This is part of a massive 6...
27. ### I Can Hyperbolic Space be affecting our view of the universe?

Usually when gravitational lensing is discussed, the examples are those of matter bending spacetime into a positive curvature. https://commons.wikimedia.org/wiki/File:Gravitational_lens-full.jpg In these cases, distortion of light is clearly evident as images of galaxies from behind these...
28. ### Arc-Length Parameterization

Homework Statement Find the arc-length parameterization for r(t)=\left< { e }^{ 2t },{ e }^{ -2t },2\sqrt { 2 } t \right> ,t\ge 0 Homework Equations s(t)=\int { \left| \dot { r } (t) \right| dt } The Attempt at a Solution \dot { r } (t)=\left< { 2e }^{ 2t },-2{ e }^{ -2t },2\sqrt { 2 }...
29. ### Find the center of the circle of curvature

Homework Statement For the curve with equation y={ x }^{ 2 } at the point (1, 1) find the curvature, the radius of curvature, the equation of the normal line, the center of the circle of curvature, and the circle of curvature. Homework Equations The Attempt at a Solution \kappa \left( 1...
30. ### Curvature using Mathematica

Homework Statement Use Mathematica to calculate the Gaussian curvature of the plane Ax+By+Cz=D, in which A, B, C, and D are constants and C≠0. Use the following data: Homework Equations The Attempt at a Solution First I found the line curvature. As here: That code gets the result...
31. ### I Exploring Curvature

Here's what I'm watching: At about 1:35:00 he leaves it to us to look at a parallel transport issue. Explicitly to caclculate ##D_s D_r T_m - D_r D_s T_m## And on the last term I'm having some difficulties, the second christoffel symbol. So we have ##D_s [ \partial_r T_m - \Gamma_{rm}^t T_t]##...
32. ### I 3D space curvature visualization

I want to program space curvature vizualizaion. I want to have an observer as a player that moves in 3d curved space and surrounding objects that will show curvature by distortion when player passes near them. I am concerned about some points: 1. What curvature to choose in order to experience...
33. ### B Gravity and relative motion

Dear PF Forum, I have a confusion about gravity. And frankly I don't know if this question belongs to this sub forum (cosmology, general physic?). Gravity attracts object - Newton Gravity curves space time - Einstein. Why we revolve around a massive object? Because that massive object curves...
34. ### B Gaussian Curvature and Riemmanian Geometry

Please bear with me because I'm only in Pre-calculus and am taking basic high school physics. This is completely outside of my realm but curiosity has taken the better of me. I just learned last week about the difference between Euclidean Geometry and Riemmanian Geometry (from another thread...

G-Waves is a buzzword recently :) At the beginning I thought G-waves as the propagation of the changes of the curvature caused by a mass when the status of the mass (e.g. value or location) changes...But moment ago, I was told that G-waves are different from the waves that transmitting the...
36. ### Curvature of a group manifold

So I'm working with some group manifolds. The part that's getting to me is the Ricci scalar I'm using to describe the curvature. I have identified the groups that I'm using but that's not really relevant at the moment. We're using a left-invariant metric ##\mathcal{M}_{ab}##. Now I've got the...
37. ### Find Tangential Component of Acceleration and Curvature

Homework Statement Homework Equations I know that the tangential accel is v = wr and that Centripetal = v^2/r The Attempt at a Solution For A, I thought it would be straight forward if I had the radius as well as omega. I know that the distance between A and B is 60in, but I don't think it...
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### Finding Curvature of Function

Homework Statement For the first problem I am asked to find the curvature for y=cosx We are studying vector value functions so I tried to rewrite this as a vector valued function so I can find the curvature. I just chose r(t)= <t,cost,0>. I found rI(t)=<1,-sint,0> and rII(t)=<0,-cost,0> and...
39. ### Deriving component form of Riemann tensor in general frame

How does one derive the general form of the Riemann tensor components when it is defined with respect to the Levi-Civita connection? I assumed it was just a "plug-in and play" situation, however I end up with extra terms that don't agree with the form I've looked up in a book. In a general...
40. ### Definition for curvature of worldline in Minkowski space

Hello, in this section of the wiki article on Rindler coordinates it is stated that the proper acceleration for an observer undergoing hyperbolic motion is just "the path curvature of the corresponding world line" and thus a nice analogy between the radii of a family of concentric circles and...
41. ### I Question about gravity and speed of light

Hi all, After reading about Einstein's theory of relativity I have few questions as follows 1. Let's say I am in a space lab which is traveling at the speed of half of the speed of light. So when I try to measure the speed of light coming from space I record it as 'c'. Time is running slow...
42. ### Gravity Bend

Dear PF Forum, I realize that there are 4 basic forces in our universe. Two of them are gravity and electromagnetic force. Electromagnetic travels at the speed of light. And it seems that Gravity travels/propagates at the speed of light also. Light is bent by gravity. What about gravity? Is...