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Results tagged with riemannian-geometry
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user 21907
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.
3
votes
Lipschitz constant of exponential map
I believe that the answer is positive. Let me consider an autonomous differential equation $\dot x=f(x)$ and let us assume that $f$ is Lipschitz-continuous. The flow $\phi(t,y)$ is defined by
$$
\dot …
- 16.2k
1
vote
Complex transport equation
Let me answer to the local solvability question in the $C^\infty$ category. Take a complex-valued vector field $Z=X+iY, X,Y$ real-valued vector fields such that $Z$ is always non-zero (a "principal-ty …
- 16.2k
5
votes
1
answer
355
views
Riemannian and symplectic structures
Let $(\mathcal M,g)$ be a smooth Riemannian manifold and $\Delta$ be the standard (positive) Laplace operator given in coordinates by the usual
$$
\Delta=-\vert g\vert^{-1/2}\partial_j(\vert g\vert^{1 …
- 16.2k
2
votes
1
answer
2k
views
Second fundamental form and embeddings
Let $\Sigma$ be a smooth hypersurface of a $d$ dimensional smooth Riemannian manifold $(\mathcal M, G)$;
we may see $G_x$ as a mapping from $T_x(\mathcal M)$ into $T_x^*(\mathcal M)$ so that
$$
\lang …
- 16.2k
2
votes
Derivations of $\chi^{\infty}(M)$ which are elliptic operator
First a classical result about elliptic operators: in dimension $\ge 3$ the order of elliptic operators is even. In dimension 2, say in $\mathbb R^2$ you have elliptic vector fields such as
$$
\bar \p …
- 16.2k
1
vote
An alternative representation of the principal symbol of the Laplace operator
The answer to the first question is negative on the Euclidean sphere $\mathbb S^2$. It is possible to prove that the Laplace operator on the sphere $\mathbb S^2$ is NOT the sum of two squares of smoot …
- 16.2k
2
votes
On elliptic operators on non-compact manifolds
Too long for a comment.
I would say that your problem is a semi-global solvability question. You will find in chapter 26 of Hörmander’s ALPDO a precise definition for that property which suits well th …
- 16.2k