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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Understanding step in deriving the formula of the curvature.

Last formula on third page of the document: Computation of $\vec{r'}(t)\times \vec{r''}(t)$ From the previous two formulas and using the properties of cross products we see that ...
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Curvature of hyperbolic surface

So from what I understand the curvature of a surface by calculated by inversing the radius of the osculating circle. But if a hyperbolic surface have a negative curvature, wouldn't that imply the ...
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Is the inside of a sphere a hyperbolical surface?

So since an elliptic surface with constant curvature would be a sphere, would a an hyperbolical surface with a constant curvature be the inside of a sphere if we were to go out from inside the sphere? ...
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The Riemannian Curvature in a solid sphere

Is the Riemannian Curvature at the centre of a solid sphere zero?
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How do I rate smoothness of discretely sampled data? (Picture!!!)

In the sense that the following curves pictured in order will be rated 98%, 80%, 40%, 5% smooth approximating by eye. My ideas: (1) If the curves all follow some general shape like a polynomial ...
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Push-forward of vector fields by local isometries

I am studying Riemannian Manifolds by John Lee, and there is this lemma: Lemma 7.2. The Riemann curvature endomorphism and curvature tensor are local isometry invariants. More precisely, if ...
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Curve with with curvature $k(s)\ge 1$ everywhere has diameter $\le 2$

Let $\alpha(s)$ be a simple closed plane curve. Define the diameter $d_\alpha$ of $\alpha(s)$ to be $$d_\alpha = \sup_{t,s\in\mathbb R} \| \alpha(s) - \alpha(t) \|.$$ Assume the curvature ...
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How to express curvature of a level set in terms of derivatives of a function?

Suppose I have a smooth function $u:\mathbb R^n\to\mathbb R$. Assume that its gradient doesn't vanish (near any point where we investigate it). Is there a list of different (intrinsic and extrinsic) ...
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Nice expression for curvature of a Kähler surface

Let $\Sigma$ be a Riemann surface with symplectic form $\omega$ and complex structure $J$, and denote by $g$ the induced metric. My question is Is there a nice expression of the Gaussian curvature ...
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Riemann Tensor in Particular Frame

I'm trying to reproduce a calculation which requires computing the Riemann tensor in a particular frame specified by some vierbein $e_a$. I have a complete expression for the spacetime metric in some ...
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Maximally symmetric manifold with boundary and non-vanishing extrinsic curvature?

I was wondering if the following requirements are compatible: Given a $d$-dimensional manifold with boundary $M$ with $\partial M\neq \emptyset$ endowed with a metric $g$. The following conditions ...
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What is meant by a “curvature-line parametrization” of a surface?

Could anyone explain to me what it means if a surface is curvature-line parametrized? What does it mean intuitively and how exactly is it different from any other parametrization? I've been looking ...
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Smooth Conjugate Net vs. Curvature-Line Parametrization

so I was wondering what a smooth conjugate net exactly is, intuitively? Also, what exactly is a curvature-line parametrization? What would it mean that a smooth conjugate net is orthogonal? Why is it ...
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Concave optimization and corner solution

I have a optimization problem as follows: Assumptions: $f$ is an increasing and convex function on $R^+$ such that: $f(x): R^+\rightarrow R^+, \quad f(0)=0, \quad f'(x)\ge1,\quad f''(x)\ge 0 ...
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Curvature of a 3D trajectory for which I know data points

In order to simulate an airplane model, I need to change its orientation knowing the curvature of its trajectory. The simulator gives me the plane position, so in order to perform my orientation ...
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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 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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Expansion of path-ordered integral and curvature

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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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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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 $S'=f(S)$. ...