Pseudo-Riemannian tensor and Morse index

[Jimmy Snyder]

Let g_{ij} be a tensor, where 0 \leq i,j \leq n. The Morse index \mu is the number of negative eigenvalues of g. On page 469 of Eberhard Zeidler's QFT III: Gauge Theory, it says that g is Riemannian if \mu = 0 and pseudo-Riemannian if 0 < \mu < n. Is this correct? If so, what kind of tensor is it when \mu = n?

 

[I like Serena]

A Riemannian manifold requires the metric tensor to be positive-definite.
Positive-definite is equivalent to all eigenvalues being positive.
If one or more is negative, the tensor is indefinite.
It can still be used for a pseudo-Riemannian manifold.

The reason I can think of for a distinction for \mu = n, is that the tensor is negative-definite.

 

[Matterwave]

I'd imagine that if you had all negative eigenvalues, you could always just pick the negative of the metric and you'd have a Riemannian manifold again. I'm not sure such a sign would be physical from a practical point of view.

From a mathematical point of view, I'm not sure what it would mean.

 

[quasar987]

Let g_{ij} be a tensor, where 0 \leq i,j \leq n. The Morse index \mu is the number of negative eigenvalues of g. On page 469 of Eberhard Zeidler's QFT III: Gauge Theory, it says that g is Riemannian if \mu = 0 and pseudo-Riemannian if 0 < \mu < n. Is this correct? If so, what kind of tensor is it when \mu = n?

Are you certain you have this straight? What you call the Morse index is usually just called the signature of g. The signature is a well-defined concept for any symmetric bilinear form on a vector space. The Morse index on the other hand, is a number associated with a critical point p of a Morse function f on a manifold M. It is defined as the signature of the Hessian of f at p. The Hessian of f at p is the bilinear form whose matrix wrt some coordinate chart is the matrix of second partial derivatives. It is a symmetric bilinear form obviously so we may speak of its signature. (check) The Morse condition on f just means that this bilinear form is nondegenerate so it can have any signature btw 0 and n.

It just seems very strange to me why anyone would call the signature of a bilinear form the Morse index as there is nothing at all "Morsy" about it afaik.