# Computing the vector field $\partial_X Y$ (a part of an exercise regarding the torus as a minimal submanifold of $S^3$)

TLDR: How does one compute $$\partial_X Y$$, where $$\partial$$ is the Levi-Civita connection on $$\mathbb{R}^n$$ and $$X,Y$$ are vector fields on $$\mathbb{R}^n$$ (or possibly only defined on a submanifold of $$\mathbb{R}^n$$)?

For some context, I'm working on the following exercise (which appears as Exercise 7.4 in these lecture notes).

For $$\theta \in (0,\pi/2)$$, define $$T_\theta^2 = \{ (e^{i\alpha}\cos\theta, e^{i\beta}\sin\theta) \,\mid\, \alpha,\beta\in \mathbb{R} \} \subset \mathbb{C}^2 \cong \mathbb{R}^4.$$ Determine the values of $$\theta$$ for which $$T_\theta^2$$ is a minimal submanifold of the 3-sphere $$S^3$$.

That a submanifold $$M^m \subset N^n$$ is minimal here means that $$\mathrm{trace}\;B = \sum_{i=1}^m B(X_i, X_i) \equiv 0,$$ where $$X_j$$ form an orthonormal frame of $$TM$$ and $$B$$ is the second fundamental form of $$M$$ in $$N$$, defined as $$B(X,Y) = (\nabla_X Y)^\perp,$$ where $$\nabla$$ is the Levi-Civita connection of the ambient manifold $$N$$, and $$\perp$$ denotes the normal component of the vector field.

After some work, I managed to show that $$\text{trace}\, B = (\langle \partial_{E_1}E_1, E_3 \rangle + \langle \partial_{E_2}E_2, E_3 \rangle) E_3,$$ where $$\partial_{X}Y$$ now denotes the Levi-Civita connection of $$\mathbb{R}^4$$, and where $$E_1 = (ie^{i\alpha},0), \quad E_2 = (0,ie^{i\beta}) \quad\text{and}\quad E_3 = (-e^{i\alpha}\sin\theta, e^{i\beta}\cos\theta).$$ The vectors $$E_1$$ and $$E_2$$ form an orthonormal frame for the tangent bundle of $$T_\theta^2$$, and $$E_3$$ spans its normal space in $$S^3$$. So the problem boils down to determining the values of $$\theta$$ for which $$\langle \partial_{E_1}E_1, E_3 \rangle + \langle \partial_{E_2}E_2, E_3 \rangle \equiv 0.$$

My question: How does one actually compute the vector fields $$\partial_X Y$$, i.e. in my case $$\partial_{E_1}E_1$$ and $$\partial_{E_2}E_2$$? As far as I know these are supposed to be directional derivatives, but I've never worked this in practice. I would guess that there should be a simple formula for doing this in Euclidean spaces.

• Do you know how to do directional derivatives of a scalar function? Just apply this component by component to the vector field. – Ted Shifrin Apr 29 at 17:20
• I do know how to do that, given that the scalar function is defined on $\mathbb{R}^n$ (in which case I would compute the gradient and dot it with the direction). I guess what confuses me in this case is that the vector fields $E_i$ are defined only on a submanifold, i.e. on the torus, so I don't really know what how to perform the computation. – MisterRiemann Apr 29 at 17:23
• Remember that if the vector field $X$ is tangent to $M$, you only need values of $Y$ on $M$ to compute $(\partial_X Y)(p) = (Y\circ\alpha)'(0)$ where $\alpha$ is a curve in $M$ with $\alpha(0) = p$ and $\alpha'(0) = X(p)$. – Ted Shifrin Apr 29 at 18:02
• Ah, that final comment helped me figure it out. I made it more difficult for myself than I should have. Thank you for helping, Ted! If you wish, you can post your comments as an answer and I will accept it. – MisterRiemann Apr 29 at 18:21