Recent content by Sajet

Thank you! Yes, I wasn't sure whether the sectional curvature is a continuous function on the unit tangent bundle.

Sorry if this question seems too trivial for this forum. A grad student at my university told me that a compact Riemannian manifold always has lower and upper curvature bounds. Is this really true? The problem seems to be that I don't fully understand the curvature tensor's continuity etc...

Thank you!

Hi! I'm trying to understand the formula for the Laplace-Beltrami Operator on a Riemannian manifold. (http://en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry#Gradient.2C_divergence.2C_Laplace.E2.80.93Beltrami_operator) Specifically, how the determinant of the metric tensor...

Ah, I didn't think of that. Thanks for all your help!

Mhh, the definition is definitely the same in my notes. It was supposed to define what exactly is meant when writing ##\frac{\partial g}{\partial t}## in the Ricci Flow-equation. Would you mind telling me what those three terms are when I compute ##\frac{d}{d t}\left(g(\nabla_X Y, Z)\right)##...

https://www.youtube.com/watch?v= Thank you! So what exactly is the difference between them? I would interpret: $$\frac{\partial}{\partial t}\left(g(\nabla_X Y, Z)\right)$$ as $$\frac{\partial}{\partial t} (t \mapsto \left(g_t(\nabla_X Y, Z)\right))$$ But then, how is the other...

Hi! I'm reading this script* and I fail to understand a rather simple calculation. I assume the problem lies in me not understanding the notation that is used, and I was unable to figure it out or find it in literature. We have a smooth family of metrics g = g_t on a Riemannian manifold...

Sorry, i didn't notice the post. In case anyone ever finds this through google or the search function, here it is: -\frac{1}{2} \Delta\|\nabla f\|^2 = \frac{1}{2}\text{tr}(\text{Hess}(\langle \nabla f, \nabla f \rangle )) = \frac{1}{2}\sum_{i=1}^n \langle \nabla_{X_i} \text{grad}\langle...

I got it now :)

Hi! I'm trying to understand a proof for the Bochner-Weitzenbock formula. I'm sorry I have to bother you with such a basic question but I've worked at this for more than an hour now, but I just don't get the very first step, i.e.: Where we are in a complete Riemannian manifold, f \in...

Thanks for all your help! I haven't figured it out completely but I get the basic idea now and I'm sure I'll get there soon.

Thanks. You're right, I made a mistake in the first part. |a|² - |b|², and thus 1, is correct. With regard to the second part, I'm not really used to complex manifolds... I thought, x and y were real since they are in the tangent space. Would it be possible to transfer the entire thing to real...

Thank you for your response. I'm sorry but I still couldn't figure this ds² notation out and I was hoping I could somehow get around it by using the other notation. So I tried the following: g_p(x, y) = \frac{<x,y>}{(1-|p|^2)^2} which would be identical to Ji's definition of ds² according...

Hi! I'm trying to give a few examples of symmetric manifolds. In the article "Introduction to Symmetric Spaces and Their Compactification" Lizhen Ji mentions the Poincaré disk as a symmetric space in the following way: D = \{z \in \mathbb C | |z| < 1\} with metric ds^2 =...