As discovered by Lichnerowicz and developed by Gromov & Lawson, spin geometry is a notable pocket in the study of Riemannian metrics of positive scalar curvature. It seems that Schoen and Yau's minimal hypersurface techniques may be more powerful, but I am not an expert.

Dajczer "Submanifolds and Isometric Immersions"

I suppose I'm missing some elementary facts. I know that the space of finitely additive signed measures is a dual space. But what is the predual of the space of countably additive signed measures?

Vladimir Arnold''s book "Mathematical Methods of Classical Mechanics" describes Levi-Civita parallel transport first along geodesics on a surface, then along curves on a surface, and then along curves in a higher-dimensional space. It may be of interest.

See section 2 of Gage and Hamilton's "The heat equation shrinking convex plane curves" as an application of Hamilton's general short-time existence result in his "Three-manifolds with positive Ricci curvature." Other examples would follow in the same way

Thank you very much for the answer. Are there any standard textbook references for the dim>2 cases?

I believe one dimension is usually excluded in conformal geometry, since any two metrics are in the same conformal class

The setting of the Raychaudhuri equation is essentially that of a one-dimensional timelike, null, or spacelike distribution on a pseudo-Riemannian manifold, so it is a little more general than the parallel hypersurface setting in that the orthogonal distribution is not required to be integrable. My understanding is that this generality is significant. There is also a corresponding formula (which I don't think has a name) which extends the full second fundamental form Riccati equation.

I don't know of anywhere where I find it comprehensible. It is essentially inside of volume 1 of Kobayashi and Nomizu's "Foundations of Differential Geometry". You could also check Eells and Sampson's article "Harmonic mappings of Riemannian manifolds" and Eells and Lemaire "A report on harmonic maps" and "Selected topics in harmonic maps", maybe Schoen and Yau "Lectures on Harmonic Maps"

It is also worth noting that the induced connection on $T^\ast M\otimes\phi^\ast TN$, which is somewhat more common, is given by $\nabla_i\omega_j^\alpha=\frac{\partial\omega_j^\alpha}{\partial x^i}-\Gamma_{ij}^p\omega_p^\alpha+\frac{\partial f^\beta}{\partial x^i}\Gamma_{\beta\gamma}^\alpha\omega_j^\gamma.$

Yes, the former for $M$ and the latter for $N$. They could be any connections on $TM\to M$ and $TN\to N$, it does not matter if they are metric-compatible or torsion-free.