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Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.

7 votes
1 answer
307 views

Fundamental domain for two Grassmannians

Let $\pi_1, \pi_2$ be two $k$-dimensional subspaces of $\mathbb R^n$. Using elements of the orthogonal group $O(n)$, how much can we simplify $\pi_1, \pi_2$? Certainly there always exists $A \in O(n)$ …
Uzu Lim's user avatar
  • 903
5 votes
2 answers
531 views

Bounds for metric in normal coordinate

Let $M$ be a Riemannian $n$-manifold and $x \in M$. For the metric tensor $g_{ij}$ of geodesic normal coordinates at $x$, there is a formula $g_{ij}(y) = \delta_{ij} + \frac13 R_{kijl} y^k y^l + O(\|y …
Uzu Lim's user avatar
  • 903
3 votes
0 answers
426 views

Taylor's theorem for embedded manifolds

For an embedded Riemannian manifold $M \subseteq \mathbb{R}^m$ and a point $x \in M$, there is a series expansion (page 8 of Monera's paper): $$\exp_x(t v) = x + t J_x(v) + \frac{t^2}{2!} Q_x(v) + \c …
Uzu Lim's user avatar
  • 903