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Maybe some physics-oriented introductions will be also helpful: https://link.springer.com/book/10.1007/978-3-031-05122-7 (Kontsevich’s Deformation Quantization and Quantum Field Theory, by Nima Moshay …

This lecture briefly describes some modern developments: https://www.youtube.com/watch?v=5SujiNyzEqE These two articles are old, but personally I find them interesting: https://www.sciencedirect.com/s …

This paper https://jeb.biologists.org/content/221/19/jeb178988 contains an experimental investigation of egg rolling. Theoretically, it seems "the relationship of egg shape to egg movement (e.g. rolli …

An attempt to answer the question "What can we do with Nash's embedding theorem?" is given in the paper http://mathlab.math.scu.edu.tw/mp/pdf/S30N35.pdf Very broad perspective on Nash's imbedding the …

I think Alfred Gray's book "Tubes" https://www.springer.com/gp/book/9783764369071 is relevant. See also https://www.sciencedirect.com/science/article/pii/0040938382900052 (Comparison theorems for the …

Already Riemann in his famous "On the Hypotheses Which Lie at the Bases of Geometry" concludes that the spaces of constant curvature are precisely those in which figures can move without distortion. H …

Interestingly enough, topological phases have an application even in acoustics. See https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4915042 (Floquet topological insulators for sound, by R. Fleury, A.B. K …

Sufficient conditions, in terms of geometric invariants of the domains, such as volumes, surface areas and curvature integrals of the boundaries of domains, under which one domain can be moved by an …

From The "Foreword to Feynman Lectures on Gravitation" by John Preskill and Kip S. Thorne: In §9.3, Feynman comments that he knows no geometrical interpretation of the Bianchi identity, and he s …

It seems a partial answer (the solution is found with some additional restrictions) is given in the paper http://link.springer.com/article/10.1007%2FBF00759087 (Determination of the metric from the cu …

Varadhan's result was extended to wider class of operators and manifolds by Molchanov: http://iopscience.iop.org/0036-0279/30/1/R01 (Diffusion processes and Riemannian geometry). See also http://arxiv …

It is widely accepted that the singularity of the Schwarzschild metric at the event horizon is purely an artifact of the coordinates and no physical singularity exists at the horizon. However, as Karl …

In fact Yang-Mills equations are elliptic modulo gauge transformations. In simple terms this can be explained as follows (credit: Jonathan Evans, http://www.homepages.ucl.ac.uk/~ucahjde/yangmills.htm) …

The Dynkin formula is somewhat cumbersome. Maybe a better choice is Goldberg's version http://projecteuclid.org/euclid.dmj/1077466673 In the commutator form Goldberg's result is reformulated in http:/ …

The second formula, but with the extra minus sign: $$\left[\nabla_i, \nabla_j \right]X^k = -R^k_{lij}X^l+T^l_{ij}\nabla_l X^k $$ can be found in http://www.worldscientific.com/worldscibooks/10.1142/3 …