There are two differing notions of "intermediate Ricci curvature" in the differential geometry literature. 

The first notion interpolates between sectional curvature and Ricci curvature. Given an orthonormal set of vectors x, y1, . . . , yk in a tangent space of a Riemannian manifold M, consider the sum of sectional curvatures sec(x, y1) + . . . + sec(x, yk). This quantity has been called "kth-intermediate Ricci curvature", "kth-Ricci curvature", "k-dimensional partial Ricci curvature", and "k-mean curvature". Notice when k = 1, this quantity is simply the sectional curvature sec(x, y1), and when k = dim M - 1, it is the Ricci curvature Ric(x,x).

The second notion interpolates between Ricci curvature and scalar curvature. Given an orthonormal set of vectors x1, x2, . . . , xk, consider the sum of Ricci curvatures Ric(x1,x1) + . . . + Ric(xk,xk). Names similar to those given above have also been used to describe this quantity. Notice when k = 1, this quantity is simply the Ricci curvature Ric(x1,x1), and when k = dim M, it is the scalar curvature.

Below, I have compiled a list of articles, books, and surveys which contain results pertaining to intermediate Ricci curvature, separated according to which of the two notions given above are featured. These works are ordered chronologically (by the date of appearance on arXiv in the case of the newer articles). I have included notes when the mention of kth-intermediate Ricci curvature in the work is not clear, and I have included links to publications, arXiv preprints, and AMS reviews where possible. Please let me know if I am missing anything.

Between sectional and Ricci curvature:

• • Richard L. Bishop and Richard J. Crittenden. Geometry of Manifolds. Academic Press, New York, 1964. [book, review] (Note: see pages 253–257)

  • Philip Hartman. Oscillation criteria for self-adjoint second-order differential systems and “principal sectional curvatures”. J. Differential Equations, 34(2):326–338, 1979. [article]

  • Gregory J. Galloway. Some results on the occurrence of compact minimal submanifolds. Manuscripta Math., 35:209–219, 1981. [article]

  • Ji-Ping Sha. p-convex Riemannian manifolds. Invent. Math., 83:437–447, 1986. [article]

  • Ji-Ping Sha. Handlebodies and p-convexity. J. Differential Geom., 25(3):353–361, 1987. [article, review]

  • Hung Hsi Wu. Manifolds of partially positive curvature. Indiana Univ. Math. J., 36(3):525–548, 1987. [article, review]

  • Zhongmin Shen. A sphere theorem for manifolds of positive Ricci curvature. Indiana Univ. Math. J., 38(1):229–233, 1989. [article, review]

  • Zhongmin Shen. Finite topological type and vanishing theorems for Riemannian manifolds. Ph.D. thesis, SUNY Stony Brook, 1990. [thesis]

  • Hobum Kim and Philippe Tondeur. Riemannian foliations on manifolds with non-negative curvature. Manuscripta Math., 74(1):39–45, 1992. [article, review]

  • Zhongmin Shen. On complete manifolds of nonnegative kth-Ricci curvature. Trans. Amer. Math. Soc., 338(1):289–310, 1993. [article, review]

  • Zhongmin Shen and Guofang Wei. Volume growth and finite topological type. Differential geometry: Riemannian geometry (Los Angeles, CA, 1990), Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, 54(3):539–549, 1993. [review]

  • Katsuei Kenmotsu and Changyu Xia.  Intersections of minimal submanifolds in Riemannian manifold with partially positive curvature. Kodai Math. J., 18(2):242–249, 1995. [article,review]

  • Katsuei Kenmotsu and Changyu Xia. Hadamard-Frankel type theorems for manifolds with partially positive curvature. Pacific J. Math., 176(1):129–139, 1996. [article, review]

  • Nany Lee. Determination of the partial positivity of the curvature in symmetric spaces. Ann. Mat. Pura Appl., 171(4):107–129, 1996. [article, review]

  • Vladimir Rovenski. Submanifolds with restrictions on q-Ricci curvature. New Developments in Differential Geometry, Budapest, Springer, 1996. [chapter, review]

  • Frederick Wilhelm. On intermediate Ricci curvature and fundamental groups. Illinois J. Math., 41(3):488–494, 1997. [article, review]

  • Changyu Xia. A generalization of the classical sphere theorem. Proc. Amer. Math. Soc., 125:255–258, 1997. [article, review]

  • Vladimir Rovenski. On the role of partial Ricci curvature in the geometry of submanifolds and foliations. Ann. Polon. Math., 68(1):61–82, 1998. [article, review]

  • Vladimir Rovenski. Partial Ricci curvature of a submanifold inside a cylinder or cone in Euclidean space. Sib. Math. J., 39(6):1195–1202, 1998. [article, review]

  • Bang-Yen Chen. Relations between Ricci curvature and shape operator for submanifolds with arbitrary codimensions. Glasg. Math. J. 41(1): 33–41, 1999. [article]

  • Yoe Itokawa and Ryoichi Kobayashi. Minimizing currents in open manifolds and the n-1 homology of nonnegatively Ricci curved manifolds. Amer. J. Math., 121(6):1253–1278, 1999. [article, review]

  • Vladimir Rovenski. Foliations, submanifolds, and mixed curvature. J. Math. Sci., 99(6):1699–1787, 2000. [survey, review]

  • Ximo Gual-Arnau and R. Masó. Mean curvature comparison for tubular hypersurfaces in symmetric spaces. Balkan J. Geom. Appl. 8(1):53–62, 2003. [article, review]

  • Peter Petersen and Frederick Wilhelm. On Frankel’s Theorem. 46(1):130–139, 2003. [article, review] (Note: See Section 5)

  • Burkhard Wilking. Torus actions on manifolds of positive sectional curvature. Acta Math., 191(2):259–297, 2003. [article, review] (Note: see Remark 2.4)

  • Fuquan Fang, Sérgio Mendonça, and Xiaochun Rong. A connectedness principle in the geometry of positive curvature. Comm. Anal. Geom., 13(4):671–695, 2005. [article, review]

  • Anand Dessai. Obstructions to positive curvature and symmetry. Adv. Math., 210(2):560–577, 2007. [article, review] (Note: see Proposition 19)

  • Xusheng Liu. The partial positivity of the curvature in Riemannian symmetric spaces. Chin. Ann. Math. Ser. B. 29(3):317–332, 2008. [article, review]

  • Juan-Ru Gu and Hong-Wei Xu. The sphere theorems for manifolds with positive scalar curvature. J. Differential Geom. 92(3):507–545, 2012. [article, review]

  • Hong-Wei Xu and Fei Ye. Differentiable sphere theorems for submanifolds of positive k-th ricci curvature. Manuscripta Math. 138:529–543, 2012. [article]

  • Vladimir Rovenski. On solutions to equations with partial Ricci curvature. J. Geom. Phys., 86:370–382, 2014. [article, review]

  • Dennis Gumaer and Frederick Wilhelm. On Jacobi field splitting theorems. Differential Geom. Appl., 37:109–119, 2014. [article, arXiv, review]

  • Luigi Verdiana and Wolfgang Ziller. Concavity and rigidity in non-negative curvature. J. Differential Geom. 97(2):349–375, 2014. [article, review] (Note: See Theorems B and D)

  • Christian Ketterer and Andrea Mondino. Sectional and intermediate Ricci curvature lower bounds via optimal transport. Adv. Math., 329:781–818, 2018. [article, arXiv, review]

  • Stephan Klaus. On different notions of higher curvature. An. S¸ tiint¸. Univ. Al. I. Cuza Ias¸i Inform. (N.S.), 2:327–341, 2018. [article]

  • Luis Guijarro and Frederick Wilhelm. Focal radius, rigidity, and lower curvature bounds. Proc. Lond. Math. Soc., 116(6):1519–1552, 2018. [article, arXiv, review]

  • Luis Guijarro and Frederick Wilhelm. Restrictions on submanifolds via focal radius bounds. Math. Res. Lett., 27(1):115–139, 2020. [article, arXiv, review]

  • Lei Ni. Liouville theorems and a Schwarz Lemma for holomorphic mappings between Kähler manifolds. Comm. Pure Appl. Math. 74(5):1100–1126, 2021. [article, arXiv]

  • Yousef K. Chahine. Volume estimates for tubes around submanifolds using integral curvature bounds. J. Geom. Anal., 30:4071–4091, 2020. [article, arXiv, review]

  • Yousef K. Chahine. Manifolds with integral and intermediate Ricci curvature bounds. Ph.D. thesis, UC Santa Barbara, 2019. [thesis]

  • Luis Guijarro and Frederick Wilhelm. A softer connectivity principle. arXiv:1812.01021, to appear in Comm. Anal. Geom. [arXiv]

  • Lawrence Mouillé. Local symmetry rank bound for positive intermediate Ricci curvatures. Geom. Dedicata, 216(23), 2022. [article, arXiv]

  • Lawrence Mouillé. Positive intermediate Ricci curvature with symmetries. Ph.D. thesis, UC Riverside, 2020. [thesis, errata]

  • Xiaobo Liu and Marco Radeschi. Polar foliations on symmetric spaces and the mean curvature flow. arXiv:2006.03945, 2020. [arXiv] (Note: See Theorem 1.3 and Proposition 5.1)

  • Manuel Amann, Peter Quast, and Masoumeh Zarei. The flavour of intermediate Ricci and homotopy when studying submanifolds of symmetric spaces. arXiv:2010.15742, 2020. [arXiv]

  • Miguel Domínguez-Vázquez, David González-Álvaro, and Lawrence Mouillé. Infinite families of manifolds of positive kth-intermediate Ricci curvature with k small. Math. Ann., 2022. [article, arXiv]

  • Vladimir Rovenski and Paweł Walczak. Extrinsic Geometry of Foliations. Progress in Mathematics, vol. 339, Birkhäuser, 2021. [book]

  • Lawrence Mouillé. Torus actions on manifolds with positive intermediate Ricci curvature, J. Lond. Math Soc., 106(4):3792–3821, 2022. [article, arXiv]

  • Otis Chodosh, Chao Li, and Douglas Stryker. Complete stable minimal hypersurfaces in positively curved 4-manifolds. arXiv:2202.07708, 2022. [arXiv]

  • Philipp Reiser and David Wraith. Intermediate Ricci curvatures and Gromov's Betti number bound. arXiv:2208.13438, 2022. [arXiv]

  • Kai-Hsiang Wang. Optimal transport approach to Michael-Simon-Sobolev inequalities in manifolds with intermediate Ricci curvature lower bounds. arXiv:2209.06796, 2022. [arXiv]

  • Philipp Reiser and David Wraith. Positive intermediate Ricci curvature on fibre bundles. arXiv:2211.14610, 2022. [arXiv]

  • Lee Kennard and Lawrence Mouillé. Positive intermediate Ricci curvature with maximal symmetry rank. J. Geom. Anal., 34(129), 2024. [article, arXiv]

  • David González-Álvaro and Masoumeh Zarei. Positive intermediate curvatures and Ricci flow. arXiv:2303.08641, 2023. [arXiv]

  • Hui Ma and Jing Wu. Sobolev inequalities in manifolds with nonnegative intermediate Ricci curvature. arXiv:2303.09285, 2023. [arXiv]

  • Jihye Lee and Fabio Ricci. The Log-Sobolev inequality for a submanifold in manifolds with asymptotic non-negative intermediate Ricci curvature. arXiv:2304.02672, 2023. [arXiv]

  • Leonardo F. Cavenaghi, Lino Grama, and Ricardo M. Martins. On the dynamics of positively curved metrics on SU(3)/T^2 under the homogeneous Ricci flow. arXiv:2305.06119, 2023. [arXiv]

  • Philipp Reiser and David Wraith. A generalization of the Perelman gluing theorem and applications, arXiv:2308.06996, 2023. [arXiv]

  • Philipp Reiser and David Wraith. Positive intermediate Ricci curvature on connected sums, arXiv:2310.02746, 2023. [arXiv]

Between Ricci and scalar curvature:

partially bounded Ricci tensor:

• Jon Wolfson. Manifolds with k-positive Ricci curvature. In R. Bielawski, K. Houston, & M. Speight (Eds.), Variational Problems in Differential Geometry (London Mathematical Society Lecture Note Series, pp. 182-201). Cambridge University Press, 2011. [article]

• Sebastian Hoelzel. Surgery stable curvature conditions, Math. Ann., 365:13–47, 2016. [article, arXiv]

• Diarmuid Crowley and David Wraith. Intermediate curvatures and highly connected manifolds, Asian J. Math., 26(3):407–454, 2022. [article, arXiv]

• Stephan Klaus. On different notions of higher curvature. An. S¸ tiint¸. Univ. Al. I. Cuza Ias¸i Inform. (N.S.), 2:327–341, 2018. [article]

• Mark Walsh and David Wraith. H-space and loop space structures for intermediate curvatures, Commun. Contemp. Math., 2022. [article, arXiv]

• Georg Frenck and Jan-Bernhard Kordaß. Spaces of positive intermediate curvature metrics, Geom. Dedicata, 214:767–800, 2021. [article, arXiv]

• David González-Álvaro and Masoumeh Zarei. Positive intermediate curvatures and Ricci flow. arXiv:2303.08641, 2023. [arXiv]

• Leonardo F. Cavenaghi, Lino Grama, and Ricardo M. Martins. On the dynamics of positively curved metrics on SU(3)/T^2 under the homogeneous Ricci flow. arXiv:2305.06119, 2023. [arXiv]

m-intermediate curvature:

• Simon Brendle, Sven Hirsch, and Florian Johne. A generalization of Geroch's conjecture. Comm. Pure. Appl. Math., 2023. [article, arXiv]

• Jianchun Chu, Kwok-Kun Kwong, and Man-Chun Lee. Rigidity on non-negative intermediate curvature. arXiv:2208.12240, 2022. [arXiv]

• Shuli Chen. A Generalization of the Geroch Conjecture with Arbitrary Ends. Math. Ann., 2023. [article, arXiv]

• Kai Xu. Dimension constraints in some problems involving intermediate curvature. arXiv:2301.02730, 2023. [arXiv]

• Mohammed Larbi Labbi. On a stratification of positive scalar curvature compact manifolds. arXiv:2301.05270, 2023. [arXiv]

• Tsz-Kiu Aaron Chow, Florian Johne, and Jingbo Wan. Preserving positive intermediate curvature. J. Geom. Anal., 33(336), 2023. [article, arxiv]