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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Calculating euler characteristic and geodesic curvature

We have the usual formula for the euler characteristic in differential geometry $$\chi = \frac{1}{4\pi}\int_{M}d^{2}\sigma g^{1/2}R + \frac{1}{2\pi}\int_{\partial M}ds k$$ where we define the ...
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Kinds of isometries preserving the curvature tensor

We talked about isometries /local isometries and linear isometries in the lecture, but unfortunately we did not say when the isometry preserves the curvature tensor or sectional curvature. First I am ...
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Gaussian curvature using Enneper-Weierstrass parametrization [closed]

How to calculate the gaussian curvature of a gyroid using Enneper-Weierstrass parameterization?
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Gaussian Curvature of $x^4+y^4+z^4=1$

Let $S=\{(x,y,z)\in \mathbf R^3 | x^4+y^4+z^4=1 \}$ . To compute the Gaussian curvature $k$ of $S$, I tried an elementary method to find $dN_p$. Let $\alpha (t) = (x(t),y(t),z(t))$ be an parametried ...
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How to prove and understand this result?

Suppose $(P,\pi, M)$ is a principal bundle with structure group $G$ and suppose $\omega \in \Omega^1(P,\mathfrak{g})$ is a connection on $P$ with curvature $\Omega = D\omega$. If $\sigma : U\subset M ...
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Slight confusion about Riemann curvature, in specific about $\nabla_{[X,Y]}$

In what follows I always use Einstein summation convention. The Riemann curvature is defined as $$ R(X,Y)Z = \nabla_{X}\nabla_{Y}Z - \nabla_{Y}\nabla_{X}Z - \nabla_{[X,Y]}Z $$ Now, I want to ...
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Are there any open topological spaces other than R3 with overall zero curvature (and asymptotic to R3 towards infinity)?

What I mean by this is as follows: Take an infinite flat manifold $\mathbb{R}^3$ with zero curvature. Then subtract out a knotted torus or linked tori. And sew them back in using Dehn surgery. (In ...
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Sphere curvature as calculated from Liouville's equation

Liouville's equation for Gauss curvature tells us, that when Riemannian metric has the form $f^2(du^2+dv^2)$, then its Gauss curvature $K$ is expressed by the following equation: ...
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Deriving a curvature tensor for a special connection

May be $e(x)$ a frame vector at the point $x \in M$ for a manifold $M$. Given the following connection equation: $d_Xe+e_{|X_+}-e_{|X_-}= d_Xe + J_Xe = 0$. Here, $d_X$ denotes the directional ...
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Notion of curvature for a volume embedded in $R^3$

This question might sound slightly vague, but please bear with me. If I have an orientable, closed, sufficiently smooth surface in $R^3$, I can define its principal curvatures, mean curvature as ...
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Curvature of a space curve: how do we get from the conceptual definition to K = |dT/ds|?

(Context, calculus 3) The definition of curvature I get: "How fast a curve is changing direction at a point" (Source) But the precise definition(s) I don't understand. My book has a particularly ...
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An inequality for absolute total curvature in Riemannian surfaces

Let be $M\subseteq \mathbb{R}^3$ a compact (Riemannian) surface and let be $K$ the gaussian curvature of $M$. I want to prove that $$ \int_{M} |K| \geq 4\pi(1+g(M))$$ where $g(M)$ is the genus of ...
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Does intrinsic mean existing regardless of some bigger space?

How is the arc-length of a regular parametrized curve in a surface $S\subset\mathbb{R}^3$ intrinsic? Let $\bf{x}\rm(u,v)$ be a parametrization of $S$. Letting $E,F,G$ denote the coefficients of the ...
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Can principal curvatures be complex numbers in a real world situation?

Can the equation for the principal curvatures, $k^2 - 2Hk + K = 0$ (where H is equal to the mean curvature and K is equal to the Gaussian curvature), ever have complex roots? In other words, where ...
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Why are points of zero Gaussian curvature called parabolic?

The sign of the Gaussian curvature can be used to classify points as elliptic, hyperbolic, and parabolic. Wikipedia has this image with example surfaces: I see how a hyperboloid surface has ...
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show that $K=E_2[\omega_{12}(E_1)]-E_1[\omega_{12}(E_2)]-\omega_{12}(E_1)^2-\omega_{12}(E_2)^2$?

$$K=E_2[\omega_{12}(E_1)]-E1[\omega_{12}(E_2)]-\omega_{12}(E_1)^2-\omega_{12}(E_2)^2$$ where $K$ is Gaussian curvature, $E_i$'s are tangent frame field on surface $M$ in $R^3$, $v[.]$ is directional ...
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Ruled surface out of lines of curvatures

I'm trying to proof the following statement: Let $c$ be a curve inside a surface element $f:U\rightarrow\mathbb{R}^3$ (i.e $c=f\circ\gamma$ where $\gamma:I\rightarrow U$). Then $c$ is a line of ...
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Gradient of second fundamental form

In the book I'm reading ("Differential Geometry Curves-Surfaces-Manifolds by Wolfgang Kuhnel") two definitions of prinicpal curvatures directions are presented: The extramum values of $II(X,X)$ ...
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Gaussian curvature distribution: embeddable?

Given a smooth, rotationally symmetric Gaussian curvature profile, $G(r)$, how can we know if it is embeddable in $R^3$? $G(r)$ is defined on $r=[0,R)$ where $r$ is the geodesic length from a fixed ...
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Connection and curve

Let $\nabla$ be a connection on a Riemannian manifold and let the differential of a curve be given by $$c'(t)=c_1'(t)\partial_1 + c_2'(t) \partial_2.$$ Now I was wondering how we define ...
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Singer & Thorpe Theorem on the curvature of 4-dimensional Einstein spaces

I recently asked a question here about the paper "The curvature of 4-dimensional Einstein spaces." I got stuck again with the last theorem (2.2), where I get completely lost. They start the proof by ...
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Apex of an Exponential Function

Is there a way of calculating where the apex of an exponential lies? There's probably a deeper / more mathematical way of explaining what this is exactly. The image hopefully demonstrates what I mean. ...
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Prove geodesics are straight lines if Riemann tensor is identically zero.

Suppose that $R^{a}_{bcd}\equiv 0$ in all of our manifold $M$ (in which we assume zero torsion). Prove that all geodesics are straight lines. I tried using Ricci's identity: ...
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Basic question: Riemannian Curvature is nondegenerate

$R(X,Y)=\nabla_X\nabla_Y-\nabla_Y\nabla_X-\nabla_{[X,Y]}$ is the curvature with respect to the Levi-Civita connection $\nabla$ of a metric $g$ on a manifold $M$. Define the Riemann curvature ...
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How to prove that the flat torus is indeed flat?

The $n$-dimensional torus can be obtained as a quotient: $T^n=\mathbb{R}^n/\mathbb{Z}^n$. As pointed out here, the standard metric on $\mathbb{R}^n$ is invariant under translation by the elements of ...
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Curvature tensors and bivectors

At the beginning of the paper "The curvature of 4-dimensional Einstein spaces", by Singer and Thorpe, the authors define the space $\mathcal{R}$ of curvature tensors of the vector space $V$ as the set ...
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39 views

Which ellipses settle to 1-point contacts within a snow-globe circle?

Suppose you have a solid ellipse with axes $a$ and $b$, $(x/a)^2 + (y/b)^2 = 1$, confined inside a unit-radius circle. You shake the circle like a snow globe, and the ellipse settles to the bottom ...
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An example of 4D Hypersurface in 3D

Number of combinations of 4 dimensions choosing 3 at a time is 4. Someone please give a description of a most elementary 4 Dimensional Hyper surface which has its four 3D intersections with ...
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Line Elements for $n$-dimensional hyperspheres

I'm currently in the process of deriving the components of the Riemann Curvature Tensor for a 3-sphere using the Cartan Equations. The line element I'm starting with is: $$ ds^2 = ...
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The relationship between Ricci and Gaussian curvatures

Why do we have that for a surface (dimension $2$) that $$\text{Ric}(X, Y) = K \langle X, Y \rangle ,$$ where $K$ is the Gaussian curvature and $X, Y$ are vector fields?
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Calculating the currvature of a tractrix

I'm trying to find the curvature of a tractrix expressed in the form $r(t)=(\sin{t},\cos{t}+ln(\tan{(\frac{t}{2}))} $. From what I've found on the Internet it appears that people arrive at the ...
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Curvature of plane curves

What is the neatest way to derive the formula for the curvature $\kappa =\frac{||y'x''-y''x'||}{(x'^2+y'^2)^{\frac{3}{2}}} $?
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Signed unit normal

I'm trying to study for one of my exams and the past papers have no solutions. I had to define the signed unit normal and the signed curvature. The signed curvature, $\kappa_s$ being such that ...
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What the curvature $2$-form really represents?

Let $(E,\pi,B)$ be a principal bundle with structure group $G$. The connection $1$-form can be thought of as a projection on the vertical part. It allows us to characterize the horizontal subspaces as ...
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39 views

Angle between slopes of a curve

I am trying to understand what the change in angle of the slope of a curve means. It is hard to explain with words so here's an image that should help. The red curve has had its derivative ...
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The curvatures of a transformed surface under a similarity transformation

Setup: Let $f:\mathbb R^3\to\mathbb R^3$ be a similarity transformation. Then $f=rA+b$ for some fixed orthogonal matrix $A$, vector $b$ and nonzero real $r$. Suppose $S$ is a surface, and $S'=f(S)$. ...
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Total curvature of an ovaloid.

I have the following exercise that I have to solve without using Gauss-Bonnet theorem. We say that a compact surface $\Sigma \subset \mathbb{R}^3$ is an ovaloid if the Gaussian curvature $K(p)>0$ ...
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How do you get the curvature tensor of the Schwarzschild Solution?

So, on the Wikipedia page on the derivation of the Schwärzschild solution , I get everything up to the part about the Ricci tensor. What were the components of the tensor that were used? Could ...
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Characterize the sphere using mean curvature.

We know the following result: if $\Sigma$ is a compact surface than $$ \int_{\Sigma}H^2 \ge 4 \pi, $$ where with $H= \frac{1}{2}(\kappa_1+\kappa_2)$ we denote the main curvature. I have to prove that ...
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Geometric interpretation of Gaussian curvature.

We have the following result from Do Carmo book of differential Geometry: "Let $p$ be a point of a surface $\Sigma$ such that the Gaussian curvature $K(p) \neq 0$, and let $V$ be a connected ...
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Does the Riemann tensor encode all information about the second derivatives of the metric?

In answer to this question I suggested the following as a motivation for the definition of the Riemann tensor: Let two $\mathcal M$ and $\mathcal N$ be two dim-$n$ Lorentzian manifolds with ...
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31 views

Showing a limit is equivalent to the curvature of a planar curve

I've been working on this limit for a long time and just don't know how to show this. I tried representing h and d as vectors, but I cannot seem to accurately describe them. What would be a good ...
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Hermite Spline curvature

I have a doubt about the Hermite Spline. Is it the interpolation with the minimum value of curvature among the possible interpolative functions between two points? Is it possible to demonstrate it? ...
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Invariance of ball size under translation on the pseudosphere

On the pseudosphere, do the volume of metric balls and area of metric spheres change with the centerpoint? (I'm asking because they don't on the sphere.)
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Find the curve. Differential Geometry.

Find a non plane, closed curve such that the plane curve with the same curvature as function of the arclength is not closed. Been thinking a lot in this problem and haven't got a clue. ¿Any ...
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Why is the Riemann curvature tensor the technical expression of curvature?

According to my textbook on general relativity (Sean Carrol's book) and differential geometry, the Reimann curvature tensor is the technical expression of curvature. What makes the tensor so special? ...
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Minimum Curvature Path

Let's say we are given a closed race track with a given and constant width. I am to implement an algorithm which finds both shortest path trajectory and minimum curvature trajectory for the car. I ...
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Question on calculating curvature of a surface given implicitly

I want to find, as an exercise, an expression for the curvature of a surface given by the zero set of a function. I reached a final expression, but when I test it for a sphere I get a non-constant ...
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A point of infinite curvature on a curve

Let $\gamma(t)$ be a $C^r$ smooth curve in the plane. Suppose $r\ge 2$, so that one can define the curvature $\kappa(t)$ at $\gamma(t)$. For example, $\kappa(t)=0$ means that the curve is kind of flat ...
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Using Principal Directions and Curvatures to Find Point On Surface

Given the principal directions (max and min), principal curvatures, and normal of a surface at point n, how would you go about looking for a point on the surface at a given vector distance from n? ...