Thank you very much.

So it's my great honor and pleasure to give this talk at the occasion of the start prize

for Boghard Wilking.

I know Boghard Wilking, I actually don't know how long we know each other, long time, many

decades.

We have met at many conference discussing positive curvature for instance and I want

to talk about one version of positive curvature.

And so let's get started.

So with a very simple case of surfaces which is classical and of course well understood

that's our warm up.

So in the case of surfaces we have a famous theorem, the Gauss-Bounet theorem, which says

if you have a compact connected surface without boundary for the moment then we have its Gauss

curvature and that was explained in Oslo Hamstead's talk this morning so that's quite convenient

so we now know what Gauss curvature is, it's a function on your surface.

And if you integrate this curvature over the surface then you get a number which is called

the Euler number of the manifold up to a factor of 2 pi and this is a topological quantity

or combinatorial quantity if you manufacture your surface out of triangles for example

then you can compute this Euler number in terms of numbers of vertices, edges and triangles

so it's a combinatorial thing.

So that says that you can give a surface different shapes, you can form it differently and curvature

will change accordingly but if you integrate it, the curvature then you will always get

something which is topological and does not depend on the precise form of the surface.

And as a trivial but important consequence we see that if we have a positively curved

surface so we have positive curvature then of course this integral must also be positive

and therefore the Euler number must be positive but there are not many surfaces with positive

Euler number namely it's the sphere which we have seen in Oslo's talk this morning

and a quotient of the sphere namely projective plane so these are the only options.

So that's what happens in two dimensions for surfaces without boundary.

Now how about manifolds or surfaces with boundary?

So let's start with a corresponding Gaspard-A formula if you have a boundary then there's

an additional boundary term in the formula and so that's the geodesic curvature of the

boundary which gets integrated along the boundary and additional contribution and if you don't

know anything about this term of course positivity of the curvature does not any longer tell

you anything about the Euler number and in fact let's start with the torus here so that's

the torus the torus does not admit positive curvature we have seen this picture this morning

and so if you take this shape of the torus then outside the curvature is positive but

inside the curvature is negative and in fact if you integrate it up then the Gaspard-A

formula will tell you that you get zero so positive and negative curvature cancel out

so if we have a torus then this does not allow positive curvature but now we remove a little

piece from the torus you see that here so we remove this little window from the torus

which leaves us with a manifold or surface with boundary question can we now put positively

curved can we put this in a positively curved form and for this I have prepared a little

animation so let's see if it works so we start deforming our torus and see in the end because

you see we end up with something which looks like a subset of the sphere which is in fact

positively curved so if we remove this little piece of the torus indeed then we can it can

be given a metric with curvature and the reason why it's no longer a contradiction to gasp

bernays because we have not assumed anything on the G dizzy curvature of the boundary right

so this tells us that if we want to have something and we want to say something about surfaces

with boundary we also need to impose conditions on the curvature of the boundary and one way to do

that would be just to assume convexity also the to assume that the curvature of the boundaries