Riemannian geometry is a difficult subject to type in LaTeX. For example, there are concepts in Riemannian geometry which are best understood by pictures that would take no more than five seconds to draw by hand but much more time to incorporate into a LaTeX file. Also, the same can be said for long computations in Riemannian geometry (e.g., the computation of the Riemann curvature tensor, the Ricci tensor, or the scalar curvature of a Riemannian manifold). Of course, there will undoubtedly be some people who can type the contents of a subject such as Riemannian geometry very efficiently in an organized manner but even for such people writing down by hand the same contents would be more efficient.
I think that writing down your notes by hand is actually a better idea than typing your notes in LaTeX. The reason is twofold. Firstly, writing down your notes forces you to think about the material before you begin writing; you cannot simply hit the "backspace" or "cut, copy, or paste" when you write down by hand and you will have to plan out that which you are going to write. Secondly, I think writing down your notes by hand is a more involved process in that your memory of the material will be better if you do this (but there are probably people who are exceptions to this rule).
Of course, the advantage of typing your notes in LaTeX is that you have a neat and organized set of notes in a readily accessible location. However, I think that the textbook itself serves this purpose; your notes should distinguish themselves from the textbook if they are to have additional value.
My advice is actually not to take down too many notes when you study. The reason is that this slows you down sufficiently that the disadantages outweigh the advantages. I think taking notes in some form is definitely useful when learning a new topic but this depends on what you actually write down. Moreover, for practical purposes it might not be possible to write down everything.
I would advise you to write down enough that you have an understanding of the "big picture", i.e., conceptualize what you are studying. If you are learning Riemannian geometry (as is the case), then this does not involve writing down every theorem and every proof. You should rather write down those definitions, results, and examples that help you to see the intuitions in the subject. For example, the existence and uniqueness of geodesics in a Riemannian manifold (subject to two initial conditions) is essentially a "clever" application of the fundamental existence and uniqueness theorem for ordinary differential equations. If you attempt to write down the entire proof (line by line), then the real reason the result is true will become obscured.
Similarly, the understanding that the proof of the existence and uniqueness of the Levi-Civita connection on a Riemannian manifold is similar in approach to the construction of the differential operator in the de Rham complex of a smooth manifold is also an important conceptual idea. You should highlight these conceptual ideas as much as possible rather than the line by line logic and computations that occur in the proof. The key to really understanding and remembering a subject is to have a solid conceptual framework in your mind.
I should add that memorizing formulas in Riemannian geometry is best aided by explicit computations involving the formula. You should try to compute at least a few examples of geodesics, the Riemann curvature tensor, the Ricci tensor, scalar curvature etc. in order to best understand the mechanisms of the formulas. The connections in your brain between the conceptual framework of the subject and the explicit computations are strengthened by doing as many computations as possible.
In summary, my advice is to write down as much as possible your intuitions in the subject and why you believe certain results are true. In fact, you should be creative in this process; if you understand the proof of a result but if you feel that you are not sure why it is true, then try to search for a deeper understanding that will help you to realize this. For example, doing examples with the theorem in question is often helpful if you really wish to understand why the theorem is true; write down these examples and your conclusions based upon them. If you find that you forget what you are learning, then I think writing down everything does not in fact help as much as it seems it should; you would be making better use of your time conceptualizing the material and thinking about the bigger picture.
I hope this helps!
