# Questions tagged [curvature]

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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### Can I assume a horizontal vector is a horizontal lift?

I am reading a proof that the curvature 2-form is given by $(F_A)_p(v,w) = dA_p(v,w) + [A_p(v),A_p(w)]$ where $A\in \Omega^1(P;\mathfrak{g})$ is a connection 1-form on $P\xrightarrow{\pi}M$. One of ...
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### Tangent developable is locally isometric to an open set in ℝ²

I'm trying to solve this problem about tangent developable from a differential geometry exam $\sf 1996\ Q3$: My working: First part $\ldots\ldots$deduce that $Σ$ is ruled (that is, each point of $Σ$ ...
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### Checking an error in a Differential Geometry problem on curvature and diffeomorphism types of compact surfaces

I am currently working through the following old exam problem. I have reached part (d)(i) but I believe the result required still is not guaranteed even with the new hypotheses. For example, if we ...
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### Gaussian curvature of Hyperbolic Plane.

I'm studying about Gaussian curvature from Elementary Differential Geometry - Barret O'Neill, Revised Second Edition. More precisely, in geometric surfaces. In section 7.2, example 2.5, it comes up ...
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### Radius relation of four consecutively touching circles $O_7,O_2,O_3,O_6$ with common tangent circles $O_1,O_4$

A construction of four consecutively touching circles $O_7,O_2,O_3,O_6$ that all touches circles $O_1,O_4$: Triangle $O_1O_2O_3$ has outer Soddy circle $O_4$, so that the circles $O_1,O_2,O_3,O_4$ are ...
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### Diagonal metric with induced metric being diagonal as well

A nondegenerate cyclic surface in $\Bbb L^3$ (Lorentz-Minkowski space) with constant (Gaussian curvature) $K \ne 0$ is a surface of revolution. Take a surface of revolution $S\subset \Bbb L^3$ with ...
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### Discrepancy in Calculating Scalar and Sectional Curvature in 2D Geometry

I've been working on a problem related to curvature in two-dimensional geometries and found a discrepancy between my calculations of scalar and sectional curvature, which seems to contradict the known ...
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### Examples of noncomplete, simply connected Riemannian manifolds with nonpositive sectional curvature

I will be giving a talk on the Cartan-Hadamard theorem next week (in the context of a learning/reading group), which states (in the form I will be presenting) that a complete simply connected ...
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### Verifying that $M$ has curvature $-1$ iff $f''=f$ when $g=dr^2+f(r)^2d\theta^2,M=[0,\infty)_r\times S_\theta^1$

I am currently at a difficult position, because I have to check some definitions/examples regarding hyperbolic surfaces, but I have not taken a proper course on Riemannian manifolds or surfaces in the ...
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### The integral of the absolute value of the Gaussian curvature of a compact surface

I want to prove the following theorem: Let $S$ be a compact surface, and $N:S\rightarrow \Bbb{S}^2$ the Gauss map, then we have $$\int_{S} |K| \,dA = \int_{\Bbb{S}^2} \#N^{-1} \,dA$$ where $K$ is ...
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### 1-7 Exercise 4 Do Carmo Differential Geometry

The question is: Let $C$ be a plane curve and let $T$ be the tangent line at a point $p \in C$. Draw a line $L$ parallel to the normal line at $p$ and at distance $d$ of $p$ (Fig. 1-36). Let $h$ be ...
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### Does the Tangent theorem for circumferences hold in the sphere $S^2$? [closed]

I am wondering if a kind of theorem of tangents from a point to a circumference holds in the sphere. The situation that I have in mind is the following. Take two geodesics $\gamma_1$ and $\gamma_2$ ...
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### Normal of curvature of curves in a surface

Let be $S=\{ (x,y,z) \in \mathbb{R}^3| z=y^2-3x^2 \}$ Determine the normal curvature at time $t=0$ of curves parametrized by arc length $\gamma: (-1,1)->S$ with $\gamma(0)=(0,0,0)$ Find two ...
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Let $S$ be a set of points $(x,y,z)\in\mathbb{R^3}$ that satisfy the equation $x^3+y^2+z^2=1$, $p_1=(0,1,0)$ and $p_2=(0,0,1)$ points in $S$ with the same gaussian curvature,prove that $f: S \to S$ ...