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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
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 …
1
vote
Hilbert's Theorem relevance to positive curvature
May be I misunderstood your question; I reformulate it as follows:
whether there exists a regular embedding of a complete surface of constant positive curvature in $R^3$ and whether this surface can …
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 …
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 …
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 …
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 …
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 …
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 …
1
vote
Riemannian metric and Volume form for $SE(n)$ and/or $E(n)$
Of course any left invariant metric with respect the group of orientation preserving euclidean isometries (I believe that it what you call SE(n) )
is a multiple of the standard Euclidean metric and …
1
vote
Tensor contraction and Covariant Derivative
It is a natural requirement and is more-ore less equivalent to the natural analog of the leibniz rule. Let us consider the following example:
$$
d g(v,u) = \nabla g(u, v)+ g(\nabla u, v)+ g(u, \nabla …
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 …
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 …
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 …
5
votes
"Famous" 2d Riemannian manifolds with non-constant curvature
My favorite 2D metrics of nonconstant curvature admitting a Killing vector field are the so-called Darboux-superintegrable metrics. Their definition is: the space of Killing tensors of degree two (i. …
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 …