Hello everybody,
the next date for Oral Exams in Geometry I is the 21.03. You can register for the exam
in the secretary’s office (Mathias Kall, MA 873). Other dates will be offered during
the semester.
Best regards,
Emanuel
Before the examns on the 21st and 26th, I will offer office hours on
Monday 18.2. 12-14 and Monday 25.2. 12-14.
Afterwards, during the semester break, there will be only office hours by appointment.
Best regards,
Emanuel
Hello everybody,
Tomorrow, Thursday the 14th, we will meet at 7 pm at “Cafe A” to have a good time together and welcome the end of the semester. Cafe A is the students cafe at the ground floor of the architecture building. http://goo.gl/maps/i7aWJ
Everybody is welcome and I’m looking forward to seeing you tomorrow,
Emanuel
Next week, Emanuel’s office hours will be shifted from Monday 4.2. 10:30-12:00 to Wednesday 6.2. 12-14.
Hello everybody,
I just realized that I was talking nonsense a week ago, claiming the projective metric would be given by
$$\cosh d_{pr} (P,Q) = \frac{1}{2} \ln \cr(Q,X,P,Y)$$.
The correct formula is
$$d_{pr} (P,Q) = \frac{1}{2} \ln \cr(Q,X,P,Y)$$.
I hope you realized that for yourself in the meantime 😉
Have a nice week-end,
Emanuel
In exercise 1, a minus sign was missing. It is to show $\langle x , p \rangle \le -1$ (instead of $\langle x , p \rangle \le 1$).
The exercise sheet has been updated accordingly
In the tutorials, we revisited that for a curve that satisfies $\langle \gamma , \gamma \rangle = c$ one always has $\langle \gamma’ , \gamma \rangle = 0$, i.e., $\gamma \perp \gamma’$ with respect to the product $\langle \cdot , \cdot \rangle$.
This means that tangent vectors $v$ to the quadric $Q = \left\{ x \mid \langle x , x \rangle = c \right\}$ at a point $p \in Q$ are characterized by $\langle p , v \rangle = 0 \Leftrightarrow v \in p^\perp$. However, the affine plane that is tangent to $Q$ at $p$ is given by $\left\{ x = p + v \mid v \in p^\perp \right\} = \left\{ x \mid \langle x , p \rangle = \langle p , p \rangle = c \right\}$.
In the tutorial we discussed doubly ruled quadrics, i.e., quadrics of signature (+,+,-,-) in $\mathbb{R}P^3$. The term “doubly ruled” expresses the fact that those quadrics contain two families of lines, where each family of lines generates the quadric. Two lines of those families are skew if they are contained in the same family and they intersect if they are contained in different families. Therefore, each point of the quadric can be described as the intersection point of two unique lines, which also span the tangent plane to the quadric at the intersection point.
It is an important fact that any three skew lines in $\mathbb{R}P^3$ determine a unique doubly ruled quadric that contains the three given lines. You may use this statement for the solution of Exercise 8.3.
Happy holidays,
Emanuel
In exercise 2 on sheet 5, the claim to be proven ($\textrm{cr}(y,p,x,q) = -1$) of course refers to the dual construction. (In the primal construction, these 4 lines are not concurrent).
Please participate in the doodle poll about changing Emanuel’s office hours to Monday.
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According to the poll, Emanuel’s office hours have been changed to Monday, 10:30 – 12, starting from Monday the 26th on. There is also the option to visit Thilo’s office hours on Tuesdays, 13-14.