This delightful, oh-so-lyrical topic was suggested on Facebook by my husband, Mark Dennehy. Why? I’ll ask him and get back to you at the end of this post. However, let us make a start, shall we?

The first starting point in researching this is Wikipedia. Your intrepid blogger goes and enters this in the search box and gets the following definition;

*A pseudo-Riemannian “metric” is a nondegenerate quadratic form on a real vector space Rn.*

Okay… that didn’t really help very much. What about a Riemannian metric? Wikipedia gives me the following;

*ARiemannian metric is a positive-definite quadratic form on a real vector space.*

Huh. I’m still in the dark here. So what is in fact a quadratic form?

*A Quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,*

*is a quadratic form in the variables x and y.*

Well, now at least we’re on more familiar ground. I recognise the style of equation given. Is it possible that we’re looking at a nomenclature for a form of mathematics that I’m already familiar with?

Lets have a look at the definition for a homogeneous polynomial. What does that tell us?

*A homogeneous polynomial is a polynomial whose nonzero terms all have the same degree*

Ok, that didn’t really help. However, do you notice the definitions are getting shorter? It is almost as if we’re getting closer to a form of definition we might just understand. This is very much a cause of hope, a cause of optimism. We’re on the case and we’re getting closer. Maybe *too* close…. Okay, stop being silly. Lets see what the definition of a polynomial brings us. Is that one smaller again?

*A polynomial is an expression of finite length constructed from variables (also called indeterminates) and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents.*

Ah for feck’s sake!

Yeah but hang on, though. This seems almost understandable and familiar. 2 + 2 would be a polynomial, according to this definition. Right, so now we’re getting somewhere. Let’s go back, and have a look at the definition above this.

*A homogeneous polynomial is a polynomial whose nonzero terms all have the same degree*

The same degree… without being cubed or squared? So our example of 2 + 2 would seem to still be valid. So what was the one above that?

*A Quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,*

*is a quadratic form in the variables x and y.*

So a Quadratic form seems to relate to the equation where the variables are squared. Okay, I’m balancing my understanding on a shaky tray, but it’s still hanging in there. So what’s the next one?

*ARiemannian metric is a positive-definite quadratic form on a real vector space.*

Crashing into a brick wall. I’m guessing that this is stating the equation they are speaking of is not a minus (i.e., -2y would not be included) and by the same logic would not be grafted on the -x\y section of the graft. Very much guess work here, though. However, there’s only one more definition to go. This is the Boss Level, the big cheese, the whole enchilada.

*A pseudo-Riemannian “metric” is a nondegenerate quadratic form on a real vector space Rn.*

Nope, I’m back to giving you blank stares. But with a burst of inspiration, I put ‘Pesudo-Riemannian’ into Google Images, and this is what you get;

Which suggests that these equations are of use in describing huge, unknown tracts of space, that they help us to understand the unknown, that they can do a universe more than my stupid efforts can perceive. *They see what we can’t.*

Sometimes I hate being so obtuse.

Still no clear response from hubby regarding why he suggested this topic. If he gives me a reason I’ll update this entry.

Fionaknowing your hubby as well as I do, I can pinpoint the approximate point in this blog where he will (in his oh-so-quiet manner) start to, how shall I put this nicely, “wibble” at your math (because, apparently it’s “math” not “maths”… no, I don’t understand his logic on that one either!)….. ya see that word “equation” right up there…. that’ll give him all sorts of grief…. I can hear him in my head ranting aimlessly about the lack of an “=” sign, so therefore it can’t be an equation, blah, blah, blah, yadda, yadda, yadda 😉

oh, btw, I’m trying to come up with something to counteract his proposed calculus for infants malarky!

clairePost authorThe reason for the lack of an = sign is that he has made me switch to a new laptop with a German keyboard that has cleverly hidden the = sign on me. I blame Merkel meself, it’s clearly a trap.

I do suspect his reasons for suggesting this topic may, just may, have something to do with the reasons you suggest. But I love my Maths equations, yes I do! *shakes ringlets, hugs stuffed toy, giggles, topples over*

Andrew Carlin“balancing my understanding on a shaky tray”: brilliant

IainI’m not sure but I think its R^n and not Rn.

Also it’s also more fun to say the pseudo-Riemannian properties can be determined by the product of the Hessian because then you think of old headless Prussian-esque knights having a math-off in a battlefield.