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Results tagged with riemannian-geometry
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user 14515
Riemannian Geometry is a subfield of Differential Geometry, which specifically studies "Riemannian Manifolds", manifolds with "Riemannian Metrics", which means that they are equipped with continuous inner products.
23
votes
Manifolds admitting flat connections
I did not understand the first question
Question 1 Are there manifolds with the property that each connection on is never flat?
Because one of course can construct, on any manifold, a connectio …
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18
votes
Is there a global obstruction for a diffeomorphism to be an isometry?
The answer is ``no'', the pointwise condition is not enough. The example exists in dimension 1 already and can be generalized and made arbitrary weird for all dimensions.
Consider a smooth functi …
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12
votes
Does positively curved sphere admit an isometric embedding as hypersurface in Euclidean space?
The answer of j.c. given prior to mine is of course correct but let me give a trivial reason why (in big dimensions) every Riemannian metric after an arbitrary small perturbation is not isometr …
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11
votes
Accepted
Perimeter of ellipse: Combination of two geometries
No, because otherwise we will have this property also for degenerate ellipses, which are intervals, which would imply that the euclidean distance between two (sufficiently close)
points is $\lambda …
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10
votes
Accepted
Riemannian metrics preserved by diffeomorphisms
The answer depends on the diffeomorphism.
Let me give two examples, both on the standard torus $\mathbb{R}^2/_{\mathbb{Z}^2}$ with coordinates $x,y$.
(Example 1:) $$\phi(x,y)= (x+ 1/2,y).$$
Fo …
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10
votes
Accepted
Smoothing of the distance function on a Riemannian manifold
You function $d_p$ is a Lipschitz function (w.r.t. to the Riemannian distance) with the Lipschitz constant $1$ and it is possibly, for every $\varepsilon>0$, to $\varepsilon$-approximate it by a smoot …
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8
votes
Accepted
What does it mean that the Hessian is proportional to the metric?
It is known (say, Y. Tashiro, Complete Riemannian manifolds and some vector fields, Trans.Amer.Math.Soc. 117(1965) 251–
275; I am not sure that Tashiro is the first who proved it and there were many …
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8
votes
Vector Fields in a Riemannian Manifold
I give a geometric explanation of the calculations of Willie, which simultaneously elaborates the suggestion of Deane.
The flow of a vector field commuting with Laplacian preserves the Laplacian an …
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8
votes
Questions on smoothness of Riemann metrics
I confirm the Anton's answer (No, and the phenomenon is essentially local), but I suggest another explanation which works for C^1 2-dimensional metrics.
We will look for a counterexample in the clas …
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7
votes
Accepted
Can the exponential map be used to define geodesics (and hence, generalisations of geodesics)?
Exponential map in your definition is closely related to the smooth family of smooth curves smoothly depending on the position such that in every point in every direction there exists precisely one …
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7
votes
Accepted
Conformal maps between two given domains
Any conformal map in dimensions $\ge 3$ is necessary a superposition of inversions and isometries (see e.g. the link suggested by Daniele Tampieri in his comment), so it takes the boundary of $D_1$ to …
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6
votes
Which surfaces admit unbounded-length simple geodesics?
Elipsoid does not posess unbounded geodesics with no self-intersection.
I do not know a conceptual explanation.
My explanation is that (due to integrability of the geodesic flow of ellipsoid) we …
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5
votes
Reference for when a metric on a four-manifold is Kahler?
I reformulate you question as follows: suppose we a given a Riemannian metric. How can one decide whether there exists a complex structure $J$ such that the metric is Kähler w.r.t. this complex struc …
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5
votes
Accepted
Length spectrum for Riemannian metrics in the projective plane
The answer is positive; in fact any smooth manifold has two nonisometric metrics with conjugate geodesic flows. A construction is in C. Croke, B. Kleiner, Conjugacy and rigidity for manifolds with a p …
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5
votes
The necessary and sufficient condition for $\textbf{global}$ conformal flatness of a n-dim (...
I add a small $\varepsilon$ to Robert's answer, which is a simple explanation and a simple example to
what he said concerning the 2 dim case. Conformal structure of signature (1,1) on the
surface …
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