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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.
4
votes
Accepted
Geodesic flows and Killing fields
If we have a Killing tensor field $K$ of type $(0,d)$, the function $$I:SM\to \mathbb{R}, \ I(v)= K(v,\dots, v) \ \ \ \ \ \ (\ast )$$ is constant along geodesic flow. This is a well-known knowled …
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7
votes
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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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4
votes
If there exists a function on a Riemannian manifold such that its Hessian matrix is the iden...
If a manifold is complete, the existence of the function $\phi$ such that $\nabla_i \nabla_j\phi = g_{ij}$ implies that the metric is flat and that in a `flat' coordinate system such that the metric …
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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 …
- 4,713
3
votes
Curvature of singular Riemannian metric
Under stronger regularity assumptions, an analog of the curvature exists in the weak sense, i.e., in the sense of generalized functions. The stronger regularity assumption is that the metric (in your …
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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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10
votes
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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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4
votes
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Applications of Hessian operator in the Riemann manifold. Simple samples $S_{2}(f)$
On the torus $T^2$ with the coordinates $x,y$ and the flat metric $g= dx^2 + dy^2$ take any function $f(x)$. Its hessian is given, after raising the index, by the (1,1)-tensor $f''(x) dx\otimes \fra …
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6
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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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3
votes
List of generic properties of Riemannian metrics
Generic metric does not admit a lot of properties some special metrics admit.
A good demonstration of this is the examples listed in the question (no multiple eigenvalues) or in the answers of
Matheus …
CommunityBot
- 1
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 …
- 16.5k
8
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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 …
CommunityBot
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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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18
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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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vote
Accepted
Symplectic reduction: from indefinite signature to Riemannian signature
Just take $R^4=C^2$ with complex
coordinates $z_1=x_1+ i y_1, z_2= x_2 + i y_2$ and the flat
metric $-dz_1 d\bar z_1 + dz_2 d \bar z_2$. It has signature (2,2). As the group of isometries take the …
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