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If $k_1,k_2,...,k_n$ are principal curvatures of a hypersurface in $\mathbb{R}^{n+1}$ then one can apparently locally parametrize the manifold as $(x_1,x_2,...,x_n,y)$ such that,

$y = \frac{1}{2}(k_1 x_1^2 +...+k_n x_n^2) + O(\vert x \vert ^3) $

I would like to know how this is proven.

I guess the coordinates are obtained by "integrating" the eigen-vectors of the shape operator whose eigen-values are the principal curvatures anyway. I suppose one has to be able to find out curves passing through a point which have as velocity at that point the principal directions.

But I have no experience of doing such a calculation on arbitrary manifolds and hence I can't see how to do this.

Further I would like to understand how for dimension 2 the principal curvatures (as defined above) become the minimum and the maximum of geodesic curvatures and also minimum and maximum of curvatures of curves obtained by intersecting the manifold with planes orthogonal to the tangent space. (in elementary surface theory the principal curvatures get defined as the maximum and the minimum of the above quantities)

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up vote 3 down vote accepted

This won't be a complete derivation, but I will post it here on the chance that it is useful. If no-one else comes along with a more complete explanation, I can try to expand on it at some later point.

Let $P$ be a point on your hypersurace in $\mathbb R^{n+1}$, and let $\Pi$ be the tangent plane to your hypersurface at $P$, interpreted as an actual plane in $\mathbb R^n$.

Change coordinates so that $P$ is the origin, and the tangent plane is given by $y = 0$. (Here $y$ is the $n+1$st coordinate of $\mathbb R^{n+1}$.) Now choose the first $n$-coordinates $x_1, \ldots, x_n$ so that they are orthogonal and point in the direction of the various principal curvatures, with principal curvatures $k_1,\ldots,k_n$.

The claim, then, is that your hypersurface is given locally at $P$ by the equation $y = (1/2)(k_1^2 x_1^2 + \cdots + k_n x_n^2) + O(x^3),$ or in other words, that the quadric $y = (1/2)(k_1^2 x_1^2 + \cdots + k_n x_n^2)$ has contact order 2 with the hypersurface at the point $P$.

This will follow more-or-less from the definition of the principal curvatures as eigenvalues of the second fundamental form: they encode the shape of your manifold at the point $P$ up to second order, and you can check that quadric we are writing down has the same principal curvatures at the origin, and so has the same shape as your manifold at the point $P$. Thus they coincide up to second order.

If you haven't actually checked this, that would be a useful exercise: take the quadric $y = (1/2)(k_1x_1^2 + \cdots + k_n x_n^2),$ and compute its second fundamental form at the origin; you should find that it is diagonal, with eigenvalues precisely $k_1,\ldots,k_n$.

As for your questions about the various different interpretations of the principal curvatures when $n = 2$, these are discussed extensively in one of the volumes of Spivak (I think the second volume; in any case, if you look in the contents you will find the place where he first starts to discuss curvature, and he talks about principal curvatures of surfaces in a lot of detail.)

Here is a brief hint: a 2x2 symmetrix matrix $A$ can be thought of as defining a conic section centered at the origin (via the equation $x^T A x = 1$), and the two eigendirections are the major and minor axes of the conic. (This is just linear algebra and basic analytic geometry.)

If you now apply this statement in the context of the second fundamental form for a surface in space, you will be able to derive that the two principal curvatures at a point correspond to maximizing and minizing the curvature of curves passing through that point.

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