SERVING THE QUANTITATIVE FINANCE COMMUNITY

Nonius
Topic Author
Posts: 5115
Joined: January 22nd, 2003, 6:48 am

### A Differential Geometric Question for all of you geometers out there.

some background.The k frame Stiefel Manifold in n space is the space of orthonormal k frames. It can be indentified with nxk matrices whose columns are orthonormal. It is also a homogeneous space and is isometric to SO(n)/SO(k). There is an obvious projection p:SO(n)->SO(n)/SO(k) which is given by taking the first k columns of a special orthogonal matrix. There is a natural point, sid, which is the projection of the identity under this map. The tangent space at sid can be identified with the subspace of the lie algebra so(n) that is orthogonal to the kernel of Tp(Id), which, in turn is the LieSubalgebra of the fixing group of Id in SO(N) (with the action on the left). The tangent space at sid can be identified with skew symmetric matrices of the form:A -BTransB 0where A is kxk skew sym and B is a (n-k)xn matrix. Denote this set of matrices by X.Question: are geodesics emanating from sid all of the form p(Exp(tU)) where U is in X and Exp is the usual exponential map defined on SO(n)?

sdw
Posts: 61
Joined: January 16th, 2003, 12:29 pm

### A Differential Geometric Question for all of you geometers out there.

A couple of (trivial) questions , nonius:1) What is 0 in "B 0"?2) Is N meant to be different from n?3) (to remind me) Are all geodesics emanating from id in SO(n) of the form exp(tY) where Y å so(n) (as SO(n) is a Lie Group)? Is SO(n)/SO(k) a Lie Group also? I guess not o/w the answer would be trivial.4) Where are the other differential geometers . . . .?

chiral3
Posts: 1722
Joined: November 11th, 2002, 7:30 pm

### A Differential Geometric Question for all of you geometers out there.

My initial thought, without out qualifying it, was that yes, the geodesics can be described by what you wrote, but that they are not the only ones. I am guessing that there are more non-trivial ones that satify everything.
Last edited by chiral3 on July 2nd, 2003, 10:00 pm, edited 1 time in total.

Nonius
Topic Author
Posts: 5115
Joined: January 22nd, 2003, 6:48 am

### A Differential Geometric Question for all of you geometers out there.

QuoteOriginally posted by: sdwA couple of (trivial) questions , nonius:1) What is 0 in "B 0"?2) Is N meant to be different from n?3) (to remind me) Are all geodesics emanating from id in SO(n) of the form exp(tY) where Y å so(n) (as SO(n) is a Lie Group)? Is SO(n)/SO(k) a Lie Group also? I guess not o/w the answer would be trivial.4) Where are the other differential geometers . . . .?1...the formatting is bad....B is an (n-k)xk submatrix. 2. N is the same as n. My mistake.3. No, SO(n)/SO(k) is a homogeneous space, not a Lie Group. It would be a Lie Group if it were a group, ie, if SO(k) were normal in SO(n)...but, it isn't.4. I think KR, Chiral, Alan, Caveny, Diffeo, Omar, and a few others would at least have a cursory understanding of the statement of my problem, maybe not the whole schmeer, but generally. Oh, and of course, there is N AKA Newton, who knows all math.

N
Posts: 2808
Joined: May 9th, 2003, 8:26 pm

### A Differential Geometric Question for all of you geometers out there.

Evil one,Either C3 or I can tell you exactly what the solution is (yes, the whole schmeer), but why??? It's sooooo much fun watching you struggle. Isn'tthis at least the forth thread on different approaches to the same basic problem? Keep this up and I'll remove your name from Newton-Nonius Theory. A hint: It's a lot easier if you use spinors.And I still can't believe that you think covariance matrixes are symmetric. Muhahahahahahahaaaaaaaaaaa

chiral3
Posts: 1722
Joined: November 11th, 2002, 7:30 pm

### A Differential Geometric Question for all of you geometers out there.

Peter G Tait (1831 - July 4, 1901) "Perhaps to the student there is no part of elementary mathematics so repulsive as is spherical geometry."Article on Quaternions Encyclopaedia Britannica (1911)

N
Posts: 2808
Joined: May 9th, 2003, 8:26 pm

### A Differential Geometric Question for all of you geometers out there.

Nonius,I really want to help, but to do so, you'll first need to get a white beachball and a heavy-duty black felt-tip marker. I suggestyou have one of your ladies do the spinning.N

Nonius
Topic Author
Posts: 5115
Joined: January 22nd, 2003, 6:48 am

### A Differential Geometric Question for all of you geometers out there.

QuoteOriginally posted by: NNonius,I really want to help, but to do so, you'll first need to get a white beachball and a heavy-duty black felt-tip marker. I suggestyou have one of your ladies do the spinning.Nthat will only work in a trivial case, but thanks Newton...time for the tab bro...then we'll do some math.

N
Posts: 2808
Joined: May 9th, 2003, 8:26 pm

### A Differential Geometric Question for all of you geometers out there.

that will only work in a trivial case, but thanks Newton...time for the tab bro...then we'll do some math. Wrong...Step 2) You must then go to the beach (late in the afternoon on a sunny data) to watch the bab.. er, I mean the ball's shadow.
Last edited by N on July 7th, 2003, 10:00 pm, edited 1 time in total.

Nonius
Topic Author
Posts: 5115
Joined: January 22nd, 2003, 6:48 am

### A Differential Geometric Question for all of you geometers out there.

ok, so I've convinced myself that geodesics are of the form Exp(tU)sId where sId is the nxk matrix with ones down the diag k times (n>k), and U is in the complement of the kernel of the projection at the identity. now, fix x different from mId in the Stiefel manifold. x=gH for some g and H=to the subroup of SO(N) that fixes the first k standard basis vectors. note that x=gsid by the commutivity of projection with left action. Also, SO(N) acts on left by isometries, so I think gExp(tU)mid are geodesics where U is again in the orth complement of the projection. question, do you agree that another description of the geodesics is Exp(tV)gsid where V is in the image of Ad(g) of the orth complement of the projection at the identity?

Nonius
Topic Author
Posts: 5115
Joined: January 22nd, 2003, 6:48 am

### A Differential Geometric Question for all of you geometers out there.

QuoteOriginally posted by: Noniusok, so I've convinced myself that geodesics are of the form Exp(tU)sId where sId is the nxk matrix with ones down the diag k times (n>k), and U is in the complement of the kernel of the projection at the identity. now, fix x different from mId in the Stiefel manifold. x=gH for some g and H=to the subroup of SO(N) that fixes the first k standard basis vectors. note that x=gsid by the commutivity of projection with left action. Also, SO(N) acts on left by isometries, so I think gExp(tU)mid are geodesics where U is again in the orth complement of the projection. question, do you agree that another description of the geodesics is Exp(tV)gsid where V is in the image of Ad(g) of the orth complement of the projection at the identity?alright, at least Chiral, help.....so, I've a smooth real valued function F on SO(N)/SO(K). Normally, I can define simply the gradient Grad(F) in the usual way....but, I want to push this up to the Lie Algebra of SO(N). In other words, I want to define GradNonius(F):SO(N)/SO(K)->U where U is an appropriate subset of the Lie Algebra of SO(N). The reason I want to do this is that, obviously, it is easy to work with stuff that lives in the Lie Algebra, even if you have to map it down to the tangent space of SO(N)/SO(K) at sid, and then move the projected vector back to the tangent space of some point in SO(N)/SO(K). I can't believe I've been stuck on this.

chiral3
Posts: 1722
Joined: November 11th, 2002, 7:30 pm

### A Differential Geometric Question for all of you geometers out there.

Are you back on exp[Ut] if N<K

N
Posts: 2808
Joined: May 9th, 2003, 8:26 pm

### A Differential Geometric Question for all of you geometers out there.

I've a smooth real valued function F on SO(N)/SO(K). Nonius,What's your idea of smooth. If it were smooth, you could use Lie algebra directly. Anyway thesolution isn't that interesting. You get what one would expect for constrained volatility, a non-symmetric matrix. Wherehave you hear that before?? I can't wait to tell you "I told you so".Have you discoved why you can't get the gradient? I'm afraid it's obvious dude. BTW. This is the exact problem that the Princeton prof from ETH didn't get right (remember my low key discussion on Banach spaces). Maybe Elan or LT can save the school's rep.Cifford Boy
Last edited by N on August 10th, 2003, 10:00 pm, edited 1 time in total.

N
Posts: 2808
Joined: May 9th, 2003, 8:26 pm

### A Differential Geometric Question for all of you geometers out there.

I don't think Nonius is up to the task of solving this simple problem, with or without C3's, Elan's, LT's or anyone else's help.He's been struggling with this same problem for at least 5 months. I say he can't solve it. Is there anyonewho thinks he can?

Nonius
Topic Author
Posts: 5115
Joined: January 22nd, 2003, 6:48 am

### A Differential Geometric Question for all of you geometers out there.

QuoteOriginally posted by: NI don't think Nonius is up to the task of solving this simple problem, with or without C3's, Elan's, LT's or anyone else's help.He's been struggling with this same problem for at least 5 months. I say he can't solve it. Is there anyonewho thinks he can?Newton, must I appeal to the Theorem of Rocks again? I really don't want to.