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Diameter rigidity for Kähler manifolds with positive bisectional curvature

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Ved Datar talks about his paper with Harish Seshadri where they prove that a Kahler manifold with positive bi-sectional curvature and maximal diameter is isometric to the complex projective space with the Fubini-Study metric.

Image courtesy of interviewee. June 8, 2024

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The speaker discusses their work in differential geometry, focusing on the study of geometric structures on smooth manifolds, particularly Kähler manifolds with positive bi-sectional curvature. They explain a general principle in geometry that objects with large positive curvature are relatively small compared to model spaces, and this leads to concepts of rigidity results in comparison geometry, where geometric objects are analyzed against model spaces like spheres or Euclidean space.

They elaborate on the importance of understanding curvature from intrinsic properties rather than just relying on external observations, giving examples like the Earth's curvature and the sum of angles in triangles on curved surfaces. The speaker introduces the concept of Riemannian manifolds, which possess additional structures allowing for the measurement of angles and lengths via dot products in tangent spaces.

The speaker then explains classical results regarding the diameter of Kähler manifolds and introduces diagnostic tools like Ricci curvature and the Riemann curvature tensor, emphasizing the complexity of higher-dimensional geometries and the challenges of deriving meaningful results using different curvature properties.

They highlight prior work by Myers and Chang establishing bounds on diameters and congruencies related to curvature, and discuss their contributions concerning conditions under which certain diameter rigidity results hold for Kähler manifolds. Notably, they mention that while analogous results for Kähler manifolds exist, there are limitations and uncertainties when establishing strict analogs, particularly relating to the use of bi-sectional curvature as opposed to Ricci curvature.

The conversation concludes with an open question centered on whether a specific construction of products of complex projective spaces yields the maximum diameter for Kähler manifolds given certain curvature conditions, with the speaker expressing curiosity about the outcome.

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00:00.205 --> 00:02.085
I broadly work in this area,

00:02.265 --> 00:05.565
called differential geometry, which is,

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which is the study essentially of geometric structures on,

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on objects called as, a smooth manifolds.

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Uh, so, so these, I mean,

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our study basically concerns, sort of subclass

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of manifolds called ka manifolds whose,

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certain notion

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of curvature called the bi sectional curvature is positive.

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And a, a general philosophy in geometry is that,

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suppose you have a, you have an object which has a very

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large positive curvature, then that object would be

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small compared to a, a, a, a model space,

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and the smallness can be measured,

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using various quantities like either the diameter or volume

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or, area and so on, right?

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So, just to give you a quick example, I mean, you,

00:56.565 --> 00:59.505
you can think of, the surface of of earth.

01:00.145 --> 01:02.705
I mean, earth, can be approximated as a sort

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of really large sphere.

01:04.565 --> 01:07.225
And if you look around yourself, I mean, the earth appears

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for all practical purposes flat, which means

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that the curvature is, is small, right?

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Uh, on the other hand, if you look at,

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let's say the surface of a, of a basketball, then

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of course the basketball appears very curved, right?

01:22.215 --> 01:25.395
And correspondingly, the diameter, the volume

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of the basket ball is small.

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So this is the general philosophy.

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Large positive curvature implies the object is small, right?

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So then, related to these are, are the,

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so-called rigidity results.

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So, so, this statement that I said,

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or this general philosophy, this comes under this branch

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of geometry called this comparison geometry,

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where under some curvature conditions,

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you're comparing the geometry of an abstract geometry object

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to a model space.

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That model space could be the spear, could be

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the euclidean space,

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or could be some other models in, in geometry.

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Um, so there are these companion reserve,

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which are called rigidity results, which basically say

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that under these curvature conditions, suppose your diameter

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or volume or size in any way is exactly optimal.

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Then these rigidity results basically tell you

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that the object has to be congruent to the model space.

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So there has to be isometric to the model space.

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So our work, which by the way, is a joint work with my,

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colleague, ha ri, is basically a, a,

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a theorem on the diameter rigidity of killer manifolds

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with positive bi sectional curvature.

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So, maybe let me talk a little bit about

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what positive curvature really means, right?

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So I give this example of, of the earth that,

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if you're looking around you, it appears flat,

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but one way to notice that the earth actually curves is

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to just go to a beach where there are lots of ships

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and sort of see ships disappear beyond the horizon, right?

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And suppose you have a lifeguard chair nearby,

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you could just climb that chair

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and suddenly the ship will reappear for a little while

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before again, it disappears beyond the horizon, right?

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Another great ways to just go

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to a really tall building, right?

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So for instance, if you go to the bush kfa,

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then it turns out that the sun actually sets, I think,

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maybe a couple of minutes late on the top floor as opposed

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to on the ground floor.

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So this of course matters.

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Uh, for instance, if you have to break your fast

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during rum sun, you need to know exactly when the sun sets,

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and then the sun actually sets slightly

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later on the top floor.

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And this is due to the curvature of the earth, right?

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Because the sun sets once it disappears beyond the horizon.

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So, so right, so, so, so,

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but I mean, for, from a mathematician's point of view,

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this is actually not quite satisfactory

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because you would like to understand curvature from a more

04:00.195 --> 04:01.515
intrinsic point of view.

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What do I mean by that? Uh, so for instance, if you're an,

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an on a basketball

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and you don't have access to the technology to, you know,

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build sort of buildings

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or something to climb bond, you would like to,

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just wander around on the surface of the ball

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and understand whether the surface is curved or not.

04:21.655 --> 04:25.235
So, one way to do this actually goes right back to

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I think, the first theorem that I ever learned in,

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in high school, which is namely the sum angle

04:31.315 --> 04:32.515
property of triangles, right?

04:32.975 --> 04:36.475
So we all know that the sum of angles of a triangle is equal

04:36.475 --> 04:39.235
to 180 degrees or pi radiance, right?

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Uh, on the other hand,

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unfortunately our school teacher didn't quite explain

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that this is actually not true on curved surfaces.

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So for instance, if you take the surface of a ball

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and, draw a triangle on it,

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so simplest way is just take two longitudes

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that are perpendicular to each other,

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and for the third side, take the equator, let's say, right?

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So this is a legitimate triangle

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because if you look at any two vertices, the,

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the edges sort of the shortest distance

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between these two points, right?

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So it's sort of the, the path that a bird

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or a airplane might fly, right?

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So, so it's a legitimate triangle.

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On the other hand, all the angles

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of the triangle are 90 degrees.

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So there's sum up to two 70 degrees

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or three five by two radiance.

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So you get something that's over pi, right?

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So there's, that's, greater than pi.

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So in fact, you can leverage this

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to define curion at a point.

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What you do is you draw these small triangles

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and compute the amount that the sum of angles

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of the triangle is above pi

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and just divide by the area of the triangle

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and let the triangle shrink to that point,

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whatever limit you get is defined to be the curvature.

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In this case, more specifically it's called the gauss

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curvature after the person who, who was one

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of the first persons to study this.

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So, so in fact again, for for a mathematician,

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you would like to abstract these things out.

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So if you, if you know this, I mean, all

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that we really needed to use was how to measure angles

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and length, so angles between curves and length of curves.

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Why length of curves? Because we have

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to decide whether something is the shortest path

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or not, right?

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So it turns out that both these, quantities,

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it's actually enough to define one particular structure on

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your manifold, which is namely, you just need a, a notion

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of dot products on tangent spaces at each

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point of your manifold.

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Once you have dot product, you can def define,

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you can measure angles between tangent vectors,

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and you can, by integrating the sort of velocity vectors

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of curves, you can also measure the lens of curves.

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So this additional structure, so you're a smooth manifold

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with this additional structure is called rumanian manifold.

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So now, this procedure that I've outlined,

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you can define the gauss curvature of a rumanian manifold

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of two dimensions in higher dimensions.

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Your manifold now has many different

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directions it can bend in.

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So the idea is

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to basically just compute take two dimensional sort

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of slices of your manifold

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and compute their gauss curvature.

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And this whole thing is packaged into this object called

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as the reman curvature tensor.

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It is a massively complicated object.

07:25.745 --> 07:27.465
So for instance, for something

07:27.465 --> 07:30.585
with dimension N if you're working on an in dimensional

07:30.625 --> 07:34.345
ninefold, your reman cova tensor will actually be described

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by roughly into the four components.

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It actually is, slightly less than into the four

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because of various symmetries involved,

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but it's about end to the four.

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Now, this is a massively complicated object.

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It's very difficult to sort of get a handle on.

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So, one way to try to get

07:52.865 --> 07:55.385
to an easier object is simply the trick

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that we use even in high school when, so for instance,

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in when we have a massive matrix of, in an n square area

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of numbers, a simpler quantity to try to understand is

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to just look at the trace of the matrix.

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So some of the properties of the matrix are reflected in

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how its trace behaves.

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So here we can carry out a similar procedure.

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We can try to take the trace of this reman cavera 10,

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that gives you a slightly less complicated object called the

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richi cavera tenor.

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And of course, one can continue this per,

08:27.545 --> 08:31.265
so the kova is roughly around 10 squared components.

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One can of course do this again, one can also take the trace

08:35.145 --> 08:38.065
of the avature and get scaler curvature.

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That's a single number returns out to the scaler curvature

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because it's a single number in higher dimensions.

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The geometry is just too complicated to be

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properly described by just a single number.

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So if you want

08:52.255 --> 08:55.055
to prove theorems using assumptions in the scaler curvature,

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you typically do not get very strong theorems

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because there's just too much flexibility.

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On the other hand, if you prove theorems,

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using some hypothesis on the curvature tensor,

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the full curvature tensor, then typically they're not

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as useful because you're,

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you have just assumed too much, right?

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So, so then it turns out that ika,

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which lies in the sweet spot,

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where you can prove very nice results.

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So that basically brings me to the really, I mean,

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now we can state our theorem essentially.

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So, there's this classical result due to Myers,

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in the 1940s, which basically said

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that if the ika voucher is positive, how positive,

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you have to always in, in geometry, in order

09:39.265 --> 09:42.925
to state theorems, you have to always rescale at some point.

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So that, right, so, so you have

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to normalize some things.

09:47.635 --> 09:52.135
So suppose the ika voucher is above, let's say N minus one,

09:52.505 --> 09:57.335
where n is the dimension of your, of your manifold.

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Then it turns out that the diameter

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of your dian manifold is bounded above by pi.

10:04.265 --> 10:07.315
Okay? And then analog isly, as I said,

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whenever you have these comparison results,

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you have analogous rigidity results.

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There was a rigidity theorem due to Chang in the 1970s,

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which says that if you have equality, then your manifold has

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to be congruent to the usual real sphere.

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Okay? So now of course, natural question is what happens

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for Kayla manifolds?

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Do you get something better?

10:30.025 --> 10:31.285
So it turns out

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that spheres are usually not Caleb manifolds except,

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for, for our usual two dimensional sphere,

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that's a K manifold.

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But the other spheres are not Caleb manifolds.

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So you have to replace your model space, in this case.

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So the correct model space turns out to be what is called

10:48.365 --> 10:50.845
as the complex predictive space,

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with the ish 2D geometry, right?

10:54.245 --> 10:56.985
So, there then a natural question is, is the version

10:56.985 --> 10:59.505
of Myers Theorem true for, for this setup?

11:00.005 --> 11:04.145
It turns out it's not true, unfortunately, but, what Li

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and Wang observed is that if you replace the richi curvature

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with another curvature quantity called this the bi sectional

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curvature, then you have this estimate, this estimate holds.

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So the diameter is bounded above by the diameter

11:15.865 --> 11:19.025
of the complex projective space with the F two geometry.

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So now what happens in the equality case,

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and that is what, we, we do basically, we talked about

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what happens if, if your diameter is maximal.

11:29.125 --> 11:31.945
Now, with rigidity results,

11:31.945 --> 11:33.345
there are also these things called

11:33.365 --> 11:36.225
as almost rigidity results, which means that if you are,

11:37.285 --> 11:40.825
if your sort of diameter volume is almost maximal,

11:41.535 --> 11:44.545
then is your geometry close enough to that

11:44.805 --> 11:45.905
of the model space?

11:46.125 --> 11:50.145
So your close enough is, is measured in terms of,

11:50.305 --> 11:52.745
a distance called as the rumor house of distance,

11:53.335 --> 11:54.335
Right? So, similarly

11:54.335 --> 11:56.605
here, also one could ask the question

11:56.635 --> 11:59.045
that suppose the diameter is almost maximal,

11:59.545 --> 12:02.965
is the geometry close to that of the FDI geometry?

12:03.585 --> 12:05.285
Uh, it turns out that,

12:05.285 --> 12:07.685
so we had in fact raised this question in a survey article

12:07.685 --> 12:11.125
that we wrote, and shortly, thereafter, I mean,

12:11.125 --> 12:13.045
I think maybe a couple of months back,

12:13.255 --> 12:15.005
there was a counter example that was found.

12:15.185 --> 12:17.885
So it's actually quite a sophisticated counter example.

12:17.985 --> 12:19.605
So it turns out that this sort

12:19.605 --> 12:22.365
of almost rigidity theorem is actually false.

12:23.185 --> 12:24.765
So that leaves, I think, maybe

12:24.765 --> 12:28.085
with one interesting question, which, so I said

12:28.085 --> 12:29.765
that the exact analog

12:29.765 --> 12:33.805
of Myers theorem in the KA context is actually false.

12:34.305 --> 12:36.845
Uh, you need, so you have to replace ika

12:37.105 --> 12:38.405
by bisexual curvature.

12:38.745 --> 12:40.645
In fact, the counter example is

12:40.645 --> 12:43.285
that if you take the ika bounded below

12:43.345 --> 12:46.525
by some positive number, then it turns out that,

12:46.785 --> 12:50.325
if you take a product of complex predictive spaces

12:50.385 --> 12:54.165
of one dimension, that turns out to have a diameter

12:54.165 --> 12:57.605
that's higher than, or that's greater than the diameter

12:57.605 --> 13:00.005
of the complex projective space of end dimensions.

13:01.195 --> 13:04.575
So natural question is, is this the maximal diameter

13:04.575 --> 13:07.255
that you can ever have for klo manifolds?

13:07.355 --> 13:09.895
So just take the product of complex projective spaces,

13:10.555 --> 13:14.295
any copies of complex projective spaces of one dimension is,

13:14.485 --> 13:16.655
does that have the maximal diameter?

13:16.955 --> 13:19.015
And this, as far as I know, is still open,

13:19.035 --> 13:21.215
and I think I would be very, curious to know

13:21.215 --> 13:22.455
what the answer is.

00:00 - I broadly work in this area, called differential geometry, which is, which is the study essentially of geometric structures on, on objects called as, a smooth manifolds. so, so these, I mean, our study basically concerns, sort of subclass

00:20 - of manifolds called ka manifolds whose, certain notion of curvature called the bi sectional curvature is positive. And a, a general philosophy in geometry is that, suppose you have a, you have an object which has a very large positive curvature, then that object would be small compared to a, a, a, a model space, and the smallness can be measured, using various quantities like either the diameter or volume or, area and so on, right?

00:54 - So, just to give you a quick example, I mean, you, you can think of, the surface of of earth. I mean, earth, can be approximated as a sort of really large sphere. And if you look around yourself, I mean, the earth appears for all practical purposes flat, which means that the curvature is, is small, right? on the other hand, if you look at, let's say the surface of a, of a basketball, then of course the basketball appears very curved, right?

01:22 - And correspondingly, the diameter, the volume of the basket ball is small. So this is the general philosophy. Large positive curvature implies the object is small, right? So then, related to these are, are the, so-called rigidity results. So, so, this statement that I said, or this general philosophy, this comes under this branch of geometry called this comparison geometry, where under some curvature conditions,

01:47 - you're comparing the geometry of an abstract geometry object to a model space. That model space could be the spear, could be the euclidean space, or could be some other models in, in geometry. so there are these companion reserve, which are called rigidity results, which basically say that under these curvature conditions, suppose your diameter or volume or size in any way is exactly optimal. Then these rigidity results basically tell you

02:20 - that the object has to be congruent to the model space. So there has to be isometric to the model space. So our work, which by the way, is a joint work with my, colleague, ha ri, is basically a, a, a theorem on the diameter rigidity of killer manifolds with positive bi sectional curvature. So, maybe let me talk a little bit about what positive curvature really means, right? So I give this example of, of the earth that, if you're looking around you, it appears flat,

02:52 - but one way to notice that the earth actually curves is to just go to a beach where there are lots of ships and sort of see ships disappear beyond the horizon, right? And suppose you have a lifeguard chair nearby, you could just climb that chair and suddenly the ship will reappear for a little while before again, it disappears beyond the horizon, right? Another great ways to just go to a really tall building, right? So for instance, if you go to the bush kfa,

03:20 - then it turns out that the sun actually sets, I think, maybe a couple of minutes late on the top floor as opposed to on the ground floor. So this of course matters. for instance, if you have to break your fast during rum sun, you need to know exactly when the sun sets, and then the sun actually sets slightly later on the top floor. And this is due to the curvature of the earth, right? Because the sun sets once it disappears beyond the horizon.

03:47 - So, so right, so, so, so, but I mean, for, from a mathematician's point of view, this is actually not quite satisfactory because you would like to understand curvature from a more intrinsic point of view. What do I mean by that? so for instance, if you're an, an on a basketball and you don't have access to the technology to, you know, build sort of buildings or something to climb bond, you would like to,

04:15 - just wander around on the surface of the ball and understand whether the surface is curved or not. So, one way to do this actually goes right back to I think, the first theorem that I ever learned in, in high school, which is namely the sum angle property of triangles, right? So we all know that the sum of angles of a triangle is equal to 180 degrees or pi radiance, right? on the other hand, unfortunately our school teacher didn't quite explain

04:45 - that this is actually not true on curved surfaces. So for instance, if you take the surface of a ball and, draw a triangle on it, so simplest way is just take two longitudes that are perpendicular to each other, and for the third side, take the equator, let's say, right? So this is a legitimate triangle because if you look at any two vertices, the, the edges sort of the shortest distance between these two points, right?

05:11 - So it's sort of the, the path that a bird or a airplane might fly, right? So, so it's a legitimate triangle. On the other hand, all the angles of the triangle are 90 degrees. So there's sum up to two 70 degrees or three five by two radiance. So you get something that's over pi, right? So there's, that's, greater than pi. So in fact, you can leverage this

05:33 - to define curion at a point. What you do is you draw these small triangles and compute the amount that the sum of angles of the triangle is above pi and just divide by the area of the triangle and let the triangle shrink to that point, whatever limit you get is defined to be the curvature. In this case, more specifically it's called the gauss curvature after the person who, who was one of the first persons to study this.

06:01 - So, so in fact again, for for a mathematician, you would like to abstract these things out. So if you, if you know this, I mean, all that we really needed to use was how to measure angles and length, so angles between curves and length of curves. Why length of curves? Because we have to decide whether something is the shortest path or not, right? So it turns out that both these, quantities, it's actually enough to define one particular structure on

06:31 - your manifold, which is namely, you just need a, a notion of dot products on tangent spaces at each point of your manifold. Once you have dot product, you can def define, you can measure angles between tangent vectors, and you can, by integrating the sort of velocity vectors of curves, you can also measure the lens of curves. So this additional structure, so you're a smooth manifold with this additional structure is called rumanian manifold. So now, this procedure that I've outlined,

07:01 - you can define the gauss curvature of a rumanian manifold of two dimensions in higher dimensions. Your manifold now has many different directions it can bend in. So the idea is to basically just compute take two dimensional sort of slices of your manifold and compute their gauss curvature. And this whole thing is packaged into this object called as the reman curvature tensor.

07:24 - It is a massively complicated object. So for instance, for something with dimension N if you're working on an in dimensional ninefold, your reman cova tensor will actually be described by roughly into the four components. It actually is, slightly less than into the four because of various symmetries involved, but it's about end to the four. Now, this is a massively complicated object. It's very difficult to sort of get a handle on.

07:49 - So, one way to try to get to an easier object is simply the trick that we use even in high school when, so for instance, in when we have a massive matrix of, in an n square area of numbers, a simpler quantity to try to understand is to just look at the trace of the matrix. So some of the properties of the matrix are reflected in how its trace behaves. So here we can carry out a similar procedure. We can try to take the trace of this reman cavera 10,

08:20 - that gives you a slightly less complicated object called the richi cavera tenor. And of course, one can continue this per, so the kova is roughly around 10 squared components. One can of course do this again, one can also take the trace of the avature and get scaler curvature. That's a single number returns out to the scaler curvature because it's a single number in higher dimensions. The geometry is just too complicated to be properly described by just a single number.

08:50 - So if you want to prove theorems using assumptions in the scaler curvature, you typically do not get very strong theorems because there's just too much flexibility. On the other hand, if you prove theorems, using some hypothesis on the curvature tensor, the full curvature tensor, then typically they're not as useful because you're, you have just assumed too much, right? So, so then it turns out that ika,

09:15 - which lies in the sweet spot, where you can prove very nice results. So that basically brings me to the really, I mean, now we can state our theorem essentially. So, there's this classical result due to Myers, in the 1940s, which basically said that if the ika voucher is positive, how positive, you have to always in, in geometry, in order to state theorems, you have to always rescale at some point. So that, right, so, so you have

09:46 - to normalize some things. So suppose the ika voucher is above, let's say N minus one, where n is the dimension of your, of your manifold. Then it turns out that the diameter of your dian manifold is bounded above by pi. Okay? And then analog isly, as I said, whenever you have these comparison results, you have analogous rigidity results. There was a rigidity theorem due to Chang in the 1970s, which says that if you have equality, then your manifold has

10:19 - to be congruent to the usual real sphere. Okay? So now of course, natural question is what happens for Kayla manifolds? Do you get something better? So it turns out that spheres are usually not Caleb manifolds except, for, for our usual two dimensional sphere, that's a K manifold. But the other spheres are not Caleb manifolds. So you have to replace your model space, in this case.

10:45 - So the correct model space turns out to be what is called as the complex predictive space, with the ish 2D geometry, right? So, there then a natural question is, is the version of Myers Theorem true for, for this setup? It turns out it's not true, unfortunately, but, what Li and Wang observed is that if you replace the richi curvature with another curvature quantity called this the bi sectional curvature, then you have this estimate, this estimate holds. So the diameter is bounded above by the diameter

11:15 - of the complex projective space with the F two geometry. So now what happens in the equality case, and that is what, we, we do basically, we talked about what happens if, if your diameter is maximal. Now, with rigidity results, there are also these things called as almost rigidity results, which means that if you are, if your sort of diameter volume is almost maximal, then is your geometry close enough to that of the model space?

11:46 - So your close enough is, is measured in terms of, a distance called as the rumor house of distance, Right? So, similarly here, also one could ask the question that suppose the diameter is almost maximal, is the geometry close to that of the FDI geometry? it turns out that, so we had in fact raised this question in a survey article that we wrote, and shortly, thereafter, I mean, I think maybe a couple of months back,

12:13 - there was a counter example that was found. So it's actually quite a sophisticated counter example. So it turns out that this sort of almost rigidity theorem is actually false. So that leaves, I think, maybe with one interesting question, which, so I said that the exact analog of Myers theorem in the KA context is actually false. you need, so you have to replace ika by bisexual curvature.

12:38 - In fact, the counter example is that if you take the ika bounded below by some positive number, then it turns out that, if you take a product of complex predictive spaces of one dimension, that turns out to have a diameter that's higher than, or that's greater than the diameter of the complex projective space of end dimensions. So natural question is, is this the maximal diameter that you can ever have for klo manifolds? So just take the product of complex projective spaces,

13:10 - any copies of complex projective spaces of one dimension is, does that have the maximal diameter? And this, as far as I know, is still open, and I think I would be very, curious to know what the answer is.

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