Instead, let me explain it like a regular person would:

Imagine walking through a park on a sunny day. You see trees all around you, some tall and others short. The ground is flat and smooth, but as you walk towards the center of the park, you notice that the grass starts to slope upwards. This creates an area with positive curvature it’s like walking in a bowl!

Now let’s say we want to measure how curvy this area really is. We can do this by taking a piece of string and wrapping it around the park, making sure that it covers all the curves and bumps. Then we stretch out the string and see how long it is this gives us an idea of how much space has been enclosed by those curves.

But what if we want to measure curvature in higher dimensions? For example, let’s say we have a 3D object that looks like a bowl or a sphere. How do we calculate its curvature without using string and measuring length? This is where Gaussian curvature comes into play!

Gaussian curvature measures the amount of bending or twisting in a surface at each point, regardless of how it’s oriented. It takes into account both principal curvatures (which are like the “high” and “low” points on a curve) and their product. This gives us an idea of whether the surface is more curved in one direction than another for example, if we have a bowl that slopes downwards towards the center, then the Gaussian curvature will be positive at every point.

But what about scalar curvature? Well, this measures the average amount of bending or twisting over an entire surface (or volume). It’s like taking all those little pieces of string we used to measure Gaussian curvature and adding them up! This gives us a global view of how curvy our object really is.

Of course, if you want more details or fancy math equations, feel free to check out some of the resources I mentioned earlier (like “Geometry, Topology and Physics” by Nakahara). But for now, let’s just enjoy the beauty of curves and bends!