Self-Osculating Surfaces: Unified Higher-Order Contact Theory

• Self-Osculating Surfaces are defined by higher-order contact phenomena that manifest in projective self-duality, Euclidean osculating constructions, and combinatorial lattice pairings.
• In projective geometry, surfaces in ℙ⁵ exhibit a four-dimensional second osculating space via contact structures and conormal maps, as illustrated by Togliatti-type and Szpond surfaces.
• Euclidean and lattice models employ osculating circles, Frenet frame analysis, and discrete pairing rules, providing insights into canal surfaces, Weingarten conditions, and growth constant estimates.

“Self-Osculating Surfaces” (SOSs) does not denote a single universally standardized object. In the literature represented here, the expression refers, or is used as a natural interpretive label, for several distinct constructions: projective surfaces $S\subset \mathbb P^5$ that are projectively equivalent to their osculating duals; projective surfaces in $\mathbb P^5$ with defective second osculating spaces of Togliatti or Shifrin type; Euclidean surfaces generated from the osculating circles or osculating planes of a space curve; and lattice surfaces in $\mathbb Z^3$ that permit noncrossing self-osculation along edges (Lvovski, 2016, Szpond, 2018, López et al., 2021, Kaya et al., 2020, Kim et al., 4 Sep 2025). The common theme is higher-order contact or “kissing” behavior, but the ambient categories, invariants, and classification problems are substantially different.

1. Terminological scope and common geometric theme

The shared vocabulary is organized around osculation: tangent data augmented by second-order or higher-order contact. In projective geometry, the relevant object is the second osculating space $\operatorname{Osc}^2_P(X)$. In Euclidean differential geometry, the basic local model is the osculating circle or osculating plane of a curve. In lattice combinatorics, self-osculation means that cells may “kiss” locally without crossing.

Context	Ambient setting	Characteristic condition
Osculating self-dual surfaces	$\mathbb P^5$	$\dim \operatorname{Osc}^2_p S = 4$ and $S$ is projectively equivalent to its osculating dual
Hypo-osculating surfaces	$\mathbb P^5$	$\dim \operatorname{Osc}^2_P(X)=4$ instead of the expected $5$
Surfaces of osculating circles / OT-ruled surfaces	$\mathbb P^5$0	Surface built from osculating circles or from rulings constrained to the osculating plane of a curve
Lattice SOSs	$\mathbb P^5$1	A surface of square faces allowing self-osculating but noncrossing edge incidences

This suggests that “SOS” is best treated as an umbrella term whose precise meaning is fixed by context. The main divide is between projective higher-order contact, Euclidean curve-generated surface theory, and combinatorial self-osculation on lattices.

2. Osculating self-dual surfaces in projective space

For a projective variety $\mathbb P^5$2, the $\mathbb P^5$3-th osculating space at a smooth point $\mathbb P^5$4 is

$\mathbb P^5$5

and for $\mathbb P^5$6 the second fundamental form satisfies

$\mathbb P^5$7

Lvovski considers $\mathbb P^5$8-dimensional subvarieties $\mathbb P^5$9 for which, at a general point, $\mathbb Z^3$0 is a hyperplane. The osculating dual $\mathbb Z^3$1 is then the closure of the set of second osculating hyperplanes, and $\mathbb Z^3$2 is osculating self-dual if some $\mathbb Z^3$3 satisfies $\mathbb Z^3$4 (Lvovski, 2016).

For surfaces, the specialization is $\mathbb Z^3$5, so $\mathbb Z^3$6 and the defining numerical condition is

$\mathbb Z^3$7

at a general point. In this setting, a contact structure on $\mathbb Z^3$8 arises from a non-degenerate skew-symmetric bilinear form $\mathbb Z^3$9 on a vector space $\operatorname{Osc}^2_P(X)$0 of dimension $\operatorname{Osc}^2_P(X)$1. A Legendrian subvariety is characterized by the isotropy of the deprojectivized tangent spaces for $\operatorname{Osc}^2_P(X)$2. The basic bound is that if $\operatorname{Osc}^2_P(X)$3 is integral for the contact structure, then

$\operatorname{Osc}^2_P(X)$4

hence $\operatorname{Osc}^2_P(X)$5. If $\operatorname{Osc}^2_P(X)$6 is Legendrian, not contained in a hyperplane, and equality holds generically, then $\operatorname{Osc}^2_P(X)$7 for general $\operatorname{Osc}^2_P(X)$8, and the symplectic identification $\operatorname{Osc}^2_P(X)$9 yields osculating self-duality.

A major source of examples is the conormal construction. If $\mathbb P^5$0 is a general hypersurface of degree $\mathbb P^5$1, its conormal variety $\mathbb P^5$2 is Legendrian, and under Bryant’s birational contact isomorphism

$\mathbb P^5$3

the image

$\mathbb P^5$4

is Legendrian, not contained in a hyperplane, satisfies $\mathbb P^5$5 for general $\mathbb P^5$6, and is osculating self-dual. For $\mathbb P^5$7, this produces surfaces in $\mathbb P^5$8, giving a large family of Legendrian SOSs parameterized by general hypersurfaces of degree $\mathbb P^5$9 in $\dim \operatorname{Osc}^2_p S = 4$0.

Lvovski also exhibits non-Legendrian osculating self-dual surfaces. For $\dim \operatorname{Osc}^2_p S = 4$1, the parametrized surface

$\dim \operatorname{Osc}^2_p S = 4$2

is osculating self-dual but not Legendrian with respect to any contact structure; it is projectively equivalent to Togliatti’s “surface (II)”. This rules out the misconception that projective self-osculation in $\dim \operatorname{Osc}^2_p S = 4$3 is exhausted by the Legendrian/contact-geometric mechanism.

3. Hypo-osculating and Togliatti-type surfaces in $\dim \operatorname{Osc}^2_p S = 4$4

A different projective use of higher-order osculation begins with the jet-evaluation map

$\dim \operatorname{Osc}^2_p S = 4$5

whose image defines the $\dim \operatorname{Osc}^2_p S = 4$6-th osculating space $\dim \operatorname{Osc}^2_p S = 4$7. For a surface $\dim \operatorname{Osc}^2_p S = 4$8 and $\dim \operatorname{Osc}^2_p S = 4$9, the expected dimension is

$S$0

If $S$1 at a general point, the surface has defective osculating behavior of order $S$2, or is hypo-osculating of order $S$3 (Szpond, 2018).

Szpond’s central example is the $S$4 surface $S$5, a rational surface of degree $S$6 obtained from the blow-up of $S$7 at the nine points of the $S$8-configuration and embedded by the six quartics

$S$9

Equivalently,

$\mathbb P^5$0

For all points of the smooth locus $\mathbb P^5$1, one has

$\mathbb P^5$2

so $\mathbb P^5$3 satisfies a single Laplace equation of order $\mathbb P^5$4. The determinant of the Hessian matrix of $\mathbb P^5$5 vanishes identically, the generic rank is $\mathbb P^5$6, and the associated second osculating hyperplane sections correspond exactly to unexpected quartic curves through the $\mathbb P^5$7-configuration. There is a $\mathbb P^5$8–$\mathbb P^5$9 correspondence between unexpected quartics and second osculating spaces of $\dim \operatorname{Osc}^2_P(X)=4$0 at smooth points; for a general point, the divisor cut out by $\dim \operatorname{Osc}^2_P(X)=4$1 is irreducible.

This behavior contrasts with the classical Togliatti and Shifrin surfaces. Togliatti’s classical surface

$\dim \operatorname{Osc}^2_P(X)=4$2

is hypo-osculating of order $\dim \operatorname{Osc}^2_P(X)=4$3, but all divisors cut out by second osculating spaces are reducible. Shifrin’s surface, the image of $\dim \operatorname{Osc}^2_P(X)=4$4 by $\dim \operatorname{Osc}^2_P(X)=4$5, is perfectly hypo-osculating in the sense that $\dim \operatorname{Osc}^2_P(X)=4$6 for every point, and again all divisors cut out by second osculating spaces are reducible. Szpond’s $\dim \operatorname{Osc}^2_P(X)=4$7 is singular, with three singular points, so it does not contradict the Piene–Tai classification of smooth perfectly hypo-osculating surfaces, but it shows that outside the smooth category one obtains defective osculation with irreducible contact divisors.

The companion $\dim \operatorname{Osc}^2_P(X)=4$8-surface

$\dim \operatorname{Osc}^2_P(X)=4$9

has image $5$0, a smooth surface of degree $5$1. It is hypo-osculating with $5$2 for all $5$3 except the nine points of $5$4, where the dimension drops to $5$5; all divisors cut out on $5$6 by second osculating spaces are reducible. This yields a sharp distinction within Togliatti-type surfaces: the same numerical defect $5$7 can coexist either with irreducible or reducible high-contact divisors.

4. Euclidean surfaces determined by osculating data of curves

In Euclidean $5$8-space, the phrase is not standard, but two papers develop surface classes that are explicitly built from the osculating data of a generating curve and are presented as natural models for what one might call SOSs (López et al., 2021, Kaya et al., 2020).

For a unit-speed curve $5$9 with Frenet frame $\mathbb P^5$00, curvature $\mathbb P^5$01, torsion $\mathbb P^5$02, and radius of curvature $\mathbb P^5$03, the surface of osculating circles is

$\mathbb P^5$04

The $\mathbb P^5$05-curves are circles in the osculating plane $\mathbb P^5$06, called parallels. The set of non-regular points consists of the generator $\mathbb P^5$07 and the set where $\mathbb P^5$08. At regular points, the parallels are lines of curvature, and when $\mathbb P^5$09 the corresponding parallel is also a geodesic. The surface carries a uniparametric family of planar lines of curvature, a property shared with surfaces of revolution and Monge surfaces.

The curvature theory is explicit. Umbilical points are characterized by

$\mathbb P^5$10

If the generator is spherical, then the surface is an open subset of a sphere. If the generator is planar, the surface is a subset of the generating plane. If a compact surface of osculating circles exists, then it is a torus. The canal-surface classification is exact: a surface of osculating circles is a canal surface if and only if the generator is a Salkowski curve, equivalently $\mathbb P^5$11, so the curvature is constant. The Weingarten classification is likewise rigid: if a surface of osculating circles is a Weingarten surface, then it is an open subset of a plane, of a sphere, or its generator is a Salkowski curve; in the latter case the surface is a linear Weingarten surface of parabolic type and satisfies

$\mathbb P^5$12

In the Salkowski case, one principal curvature is

$\mathbb P^5$13

so the curvature of the generating curve appears directly as a principal curvature of the surface.

A related but distinct Euclidean class is the osculating-type ruled surface

$\mathbb P^5$14

where the ruling direction is constrained to the osculating plane of $\mathbb P^5$15. The adapted OT-frame is $\mathbb P^5$16, with

$\mathbb P^5$17

and the OT-curvatures are

$\mathbb P^5$18

The Gaussian curvature is

$\mathbb P^5$19

with $\mathbb P^5$20 and $\mathbb P^5$21. Hence $\mathbb P^5$22. The surface is developable if and only if it is a plane or the tangent surface $\mathbb P^5$23. Minimality is characterized by

$\mathbb P^5$24

The rulings are always asymptotic curves and geodesics, and they are lines of curvature exactly when $\mathbb P^5$25, that is, when the surface is tangent or the base curve is planar. These constructions reinforce a common Euclidean theme: a surface can be encoded by the second-order geometry of an underlying curve, but only under restrictive differential conditions does this yield canal, Weingarten, developable, or minimal subclasses.

5. Self-osculating surfaces on the cubic lattice

In lattice combinatorics, self-osculating surfaces are defined in a genuinely discrete sense. The setting is the $\mathbb P^5$26-dimensional hypercubic lattice $\mathbb P^5$27, with cells encoded by center coordinates. A self-avoiding $\mathbb P^5$28-manifold is a finite connected set of $\mathbb P^5$29-faces such that each $\mathbb P^5$30-face is incident to at most two $\mathbb P^5$31-faces. A self-osculating $\mathbb P^5$32-manifold (SOM) relaxes that constraint: more than two $\mathbb P^5$33-faces may meet along a $\mathbb P^5$34-face, but only with a pairing structure satisfying a local noncrossing osculating condition (Kim et al., 4 Sep 2025).

For $\mathbb P^5$35, SOMs are self-osculating surfaces in $\mathbb P^5$36. Concretely, an SOS is obtained from a fixed polyominoid (XD), that is, a connected finite set of square faces in $\mathbb P^5$37, by replacing each edge shared by more than two faces with an osculating edge and choosing a pairing among incident faces satisfying the $\mathbb P^5$38-dimensional osculating condition. Different choices at the same XD can produce different connected SOSs. The inclusion relations are

$\mathbb P^5$39

so self-osculating surfaces strictly enlarge the self-avoiding class by allowing local “kissing” without crossings.

The main asymptotic result is the existence of growth constants. For each $\mathbb P^5$40, the growth constants of closed and open SAMs, SOMs, and XDs exist. The proof combines uniform exponential upper bounds with a concatenation theorem producing the pseudo-supermultiplicative inequality

$\mathbb P^5$41

For SOMs one has the general upper bound

$\mathbb P^5$42

Specializing to $\mathbb P^5$43 gives

$\mathbb P^5$44

For the self-avoiding subset, the same paper improves the cubic-lattice bound by a $\mathbb P^5$45-dimensional twig method: $\mathbb P^5$46 Hence

$\mathbb P^5$47

The paper does not yet extend the twig refinement from SASs to SOSs, so $\mathbb P^5$48 remains the best rigorous upper bound it provides for cubic-lattice SOSs.

6. Structural distinctions, misconceptions, and open directions

The various meanings of SOS differ not only in language but in the role played by the second osculating structure. In Lvovski’s projective theory, the condition $\mathbb P^5$49 in $\mathbb P^5$50 is the prerequisite for defining an osculating dual surface and asking for projective self-duality. In Szpond’s theory, the same numerical value $\mathbb P^5$51 signals defect, because a smooth surface in $\mathbb P^5$52 is expected to have $\mathbb P^5$53. This suggests that the same osculating dimension can encode either a self-duality mechanism or a Laplace-type deficiency, depending on the ambient framework.

Several common misconceptions are excluded by the examples already available. First, not every projective SOS is Legendrian: the surface

$\mathbb P^5$54

is osculating self-dual but not Legendrian. Second, defective second osculating behavior does not force reducible contact divisors: $\mathbb P^5$55 has irreducible osculating hyperplane sections at a general point, unlike the classical Togliatti and Shifrin surfaces. Third, Euclidean surfaces built from osculating circles are not generically canal or Weingarten; those properties occur precisely under the Salkowski condition. Fourth, lattice SOSs are not merely a restatement of lattice SASs; they form a strictly larger class obtained by admitting controlled self-osculation.

The open problems are correspondingly context-dependent. In the Togliatti-type projective setting, two questions are explicit: whether companion varieties exist for all Togliatti-type varieties, and whether there exists a smooth Togliatti-type surface $\mathbb P^5$56 such that the divisor cut out by the second osculating space at a general point is irreducible (Szpond, 2018). In the lattice setting, the main unresolved directions are sharper lower bounds for SOS/SOM growth constants, extension of the twig method to SOSs and fixed polyominoids, systematic treatment of closed SOSs, and questions about critical exponents and universality (Kim et al., 4 Sep 2025). In the Euclidean setting, the explicit formulas for surfaces of osculating circles and OT-ruled surfaces suggest inverse problems and generalizations, but the cited papers formulate those as interpretive possibilities rather than a closed classification program.

Taken together, these literatures show that self-osculation is not a single theory but a family of higher-contact phenomena. In projective geometry it organizes self-duality and Laplace equations; in Euclidean differential geometry it produces surfaces encoded by the Frenet data of a curve; and in lattice combinatorics it enlarges self-avoidance by permitting noncrossing local contacts. The term “Self-Osculating Surfaces” is therefore most precise when accompanied by its ambient category and its governing notion of osculation.